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Part 8 - Examples of Good and Bad Writing. Learning to dissertation services write often works best by example. The following are excerpts from nine first-year student essays. Most of the analytical essay, examples are bad, although I did find a two good examples in the bunch. Dissertation. In most cases, the names and dates from the essays have been changed to not compromise the subject matter for future students (in other words, don't use any of the apparent research information here in your papers). I have tried to categorize the errors as best as I could. Errors or bad portions are usually bolded to help you identify them. Smith was a religious, Christian man.
His notion of monads included contextual references to dissertation God. He believed that God controls the harmony of life through these monads. The essay then goes on to discuss these monads in a Christian context. Dissertation Writing Services In Pakistan. Had the prescribing of controlled in, student omitted the above sentences, however, the discussion of religion would have been completely out of dissertation services, place, given the essay's topic. But since the essay on hector from, person being discussed had religious views that affected his theories and work, it is relevant to mention the religious aspect. Had Smith's religion not been a direct influence on his work, it would have been irrelevant. Similarly, you wouldn't mention other things about someone in dissertation an essay if it wasn't relevant to the topic. For example, it is irrelevant to mention a scientist's race in an essay about their discovery unless the race impacted the discovery. An example of this might be if a black scientist's prime motivation to find a cure for cover letter ideas, sickle cell anemia was because that disease strikes black people in proportionally higher numbers. If the same scientist was researching some aspect of services in pakistan, physics, it would probably not be relevant to mention the race at all. An introductory paragraph:
On March 4, 1849, John Smith was born to of sylvia plath Anna Bradcock Smith and James Smith. Although certainly not of humble origins, John was acquainted with several prominent and influential men of politics with whom he discussed matters of mathematics, history, science, logic, law, and theology. Smith was brilliant in each of these fields, but he became known particularly for dissertation writing in pakistan, his contributions in the fields of philosophy, mathematics, and logistics. This paper will not only shed light on some of Smith's theories and words regarding these three areas, but will also tell of the and contrat, events in his life that made him the man that he was. This is the introduction to a chronologically-ordered essay about writing services Smith's life and discoveries. As such, the choice to begin with his date of and contrat essays, birth is a good one.
The paragraph summarizes the fields touched by Smith and also mentions the key areas he studied. Dissertation Services In Pakistan. The paper sets up an expectation for essay from, the reader of both a detailed explanation of dissertation services in pakistan, Smith's discoveries and anecdotes describing his personality. The sentence structure is grammatically sound and flows well. In the late 1650's , Smith's mother returned to university London , she then pulled him out of school with the intent to dissertation services in pakistan make him a farmer . Apostrophes indicate possessiveness or contractions, not plurality. The decade is the 1650s. The sentence is a run-on. Essays. It should either end after London, beginning a new sentence with She then, or the she then should be changed to and.
To make someone a farmer is to create a farmer for them. The student meant: to turn him into dissertation writing services in pakistan a farmer or to encourage him to be a farmer. Smith invented the widgetiscope and paved the way for analytical, future widget watching. All-the-while remaining a simple and writing humble man who considered himself to be part of a team working for and contrat essays, the greater good. The bolded part is services not a complete sentence. The entire thing should be one sentence.
All-the-while does not require hyphenation. The two differing approaches of development already described, eventually led to the development of the two original branches of widgetry; fingleish and fnordleish. This sentence is analytical mispunctuated. The comma is confusing and should be removed, and the semicolon should be a colon. Another of Smith's ideas was the method of differentiation. The university re-opened after the plague in 1667. Smith was elected to a minor fellowship, and awarded a major fellowship after he received his Master's Degree (Bogus 4). After the realization that Calculus was important, and was being recognized, a document to record all of the theories became a necessity.
The Methodis Differantium, the document that contained the elements of the theory of dissertation in pakistan, differentiation, was created in 1667. Smith believed he was being pulled in two directions when it came to publishing his theories and making his work known. He felt a need for fame and analytical essay rubrics fortune, yet on the other hand he had an abundant fear of rejection. To the dismay of many future mathematicians, it was never published because of dissertation writing, Smith's fear of criticism. Since he was not focusing on publishing his work, Smith pursued his career as a professor. This so-called paragraph is an utter mess. I Like Of Sylvia Plath Essay. There are far too many ideas in it, all of which are strung together haphazardly without any logical flow. Writing Services. I'll try to and contrat essays dissect and rewrite it, but I won't make errors bold because the dissertation services in pakistan, entire paragraph would be bold if I did. First, let's pick out the different topics being addressed: the method of differentiation the university re-opening after the plague Smith's ascension through the compare and contrat, university ranks the need of a document detailing differentiation, which was eventually created Smith's mental state, desires and fears.
Now, if we replace each sentence with the number of the corresponding idea, we can see what a jumbled mess this is: 1, 2, 3, 4, 4, 5, 5, 4, 3. Don't introduce a paragraph with one topic and then leap to dissertation writing in pakistan another topic in the next sentence. Compare. While it may sometimes be necessary to mention something as an aside to complement the topic, the return to the topic should be swift and easy to understand. Don't bounce around within the paragraph as this student has done. Another problem: there doesn't seem to dissertation services be a coherent timeline within the paragraph. Did the university re-open in 1667, or was the plague in 1667? Is the student saying that Smith was elected to a minor fellowship that year or another year? Similarly, when did the major fellowship and Master's Degree come in? It's unlikely to have all happened in one year, though it is designer letter ideas possible. The document was created in 1667, it seems, but when did Smith decide not to publish and seek work as a professor instead? Also 1667?
It sounds like that was a very busy year for poor Smith! The sentences themselves are also awkwardly constructed, making the entire thing hard to understand. I'll make some assumptions regarding the services in pakistan, confusing date information. Here is how this information should have been presented: Smith's ideas on the method of plath essay, differentiation were gaining recognition in the mathematical community, which made it necessary for him to produce a document detailing all of services in pakistan, his theories on the subject. Thus, when the university re-opened in 1667 following the plague and Smith was elected to a minor fellowship, he wrote Methodis Differantium. Although Smith wished to attain fame and essay on hector from fortune, he also feared rejection. This dichotomy resulted in his failure to dissertation publish Methodis Differantium; a failure that would be mourned by mathematicians well into the future. Still, Smith was awarded a major fellowship after receiving his Master's Degree in editor phoenix [insert year]. Dissertation In Pakistan. Since he was not interested in publishing his work, he concentrated instead on pursuing a position as a professor.
Queen Esmerelda knighted Jones in 1705 to be given the title of Sir Joe Smith, which made him the essay, first scientist to be so honored for his work (Bogus) . The phrase to be given is awkward here. It would be better written: Queen Esmerelda knighted Jones in dissertation in pakistan 1705, which gave him the title of designer letter, Sir. Who else could be honoured for Smith's work other than Smith? It should say: . Dissertation Writing Services. which made him the first man to be honored for scientific work. There probably should be a page number listed in essay the citation. Jones had a main idea of analytic geometry.
What does this mean? Does the student mean that one of dissertation writing services, Jones' main ideas concerned analytic geometry? Does he mean that one of the main ideas of analytic geometry was conceived by plath Jones? Or does he mean something else entirely? This makes little sense and in pakistan is very awkward. Whether Smith made no use of the the poetry of sylvia plath essay, manuscript from which he had copied abstracts , or whether he had previously invented the widgetiscope, are questions on dissertation, which at this distance of time no direct evidence is available . If Smith made no use of the manuscript, he can't have used it to copy abstracts. This is a very awkward way of saying that the events in question happened so long ago that there is no longer sufficient evidence to answer certain questions.
It would be better written: Questions as to whether Smith made further use of the manuscript from essays, which he copied abstracts or whether he had previously invented the dissertation writing services in pakistan, widgetiscope are rooted so far in the past that it is impossible to gather sufficient direct evidence to provide answers. This is still a bit awkward. It's best when broken up into essay the iliad smaller sentences: There are still questions as to whether Smith made further use of the manuscript from which he copied abstracts or whether he had previously invented the widgetiscope. Such questions are rooted so far in the past, however, that it is impossible to gather sufficient direct evidence to provide answers. Smith formed a political plan to dissertation writing services try to persuade the Germans to attack the French due to him not agreeing with their political agendas and this proved the means of his visiting Hamburg. Due to him not agreeing with is a very awkward way of saying: because he disagreed with. The second bolded part should be a separate sentence. Proved the means of analytical essay rubrics, his visiting is a very awkward way of in pakistan, saying is why he visited. Jones explained ideas too enormous to understand, and in simplified problems too complex to approach.
Not only is this hyperbole, it's also logically impossible. If the ideas were too complicated to dissertation in pakistan understand, Jones couldn't have understood them himself. If the problems were too complex to approach, Jones could not have approached them. Mismatched Words, Phrases, and essay Pronouns. After marrying Elizabeth, Smith's father fell ill for several months.
After no sign of in pakistan, recovery, a lawyer was summoned to the manor. A will was drawn up, including one hundred acres of land, the manor house, livestock, grain, and Smith Senior's death (Bogus 10). His mother gave birth to Smith three months after Smith senior died. He was premature after suffering from illness due to the shock of compare and contrat, her husband's passing during the fall . The phrase after no sign of recovery is not properly attached to writing in pakistan Smith's father. Instead, it is saying that the designer cover letter ideas, lawyer did not recover from something.
A will does not include land, a house, etc. It states to whom such things are bequeathed. This should say: A will was drawn up leaving one hundred acres of land, the manor house, livestock and grain to [whomever]. I don't even understand how and Smith Senior's death fits into this sentence. His in the sentence His mother gave birth. refers to the antecedent Smith Senior. Thus, Smith Senior's mother gave birth to dissertation writing in pakistan Smith Senior's son. That would necessitate incest, and is clearly not what the student meant to say.
They should have simply said Elizabeth gave birth. Who else but someone's mother gives birth to them anyway? Given the confusions regarding the various Smiths, it would have been better if the student had used first names during this part of the essay. There is inconsistency in capitalization. It is from the iliad Smith Senior once, and Smith senior another time. The he in he was premature again refers to the wrong antecedent. Smith Senior was not premature. Smith did not suffer illness due to the shock of Smith Senior's passing. Elizabeth did.
This sentence says that Smith suffered the illness. The student suddenly introduces the phrase during the fall when no other mention of the season has been made. This could be confused with Smith Senior dying from a fall. Lastly, the inverse relationship between area and the tangent were never attained. The relationship is singular, even though it refers to multiple elements. In Pakistan. Thus, the verb were should be singular as well, and changed to was. It was this century where many of the essays, worlds most honorable and highly respected mathematicians created what we know today as calculus. A century is not a place, it is a section of time.
Say it is a place where. Dissertation. or a time when. In. In this case, It was this century when. Dissertation. Adding an 's' without an prescribing apostrophe in writing services this case is pluralization, not indicative of possession. The student means world's. But perhaps the essay from, largest obstacle , which the Greeks could not overcome, were their insufficient number and measuring system . Were is plural, but obstacle and system are singular. It should be was. Tragically at the age of six, Smith's father died. This says that Smith's father died at the age of six. The student means: Tragically, when Smith was six years old his father died. Jones, now familiar with Smith's discoveries, wrote Smith a letter soon after the publication of his discoveries.
After the publication of whose discoveries: Jones' or Smith's? Jones reasoned that if he could calculate the writing services in pakistan, angles of the projected colour, a new law of refraction could be made . People can make legal laws, but natural or scientific laws are discovered. To make a new law of refraction, Jones would have to alter physics. During the designer letter, seventeenth century, the dissertation writing, inhabitants of England did not realize the importance of scientific advancement. Inhabitants could well mean non-human creatures, and cover letter is thus a poor choice of a word. Are we to understand that ALL of the people in England failed to dissertation realize the importance of scientific advancement for an entire century? It would have been better if the student had said most people in England. At the current time, the dominant belief was that light traveled in wave . The current time is the cover ideas, moment the reader is reading the sentence. The student meant to say that the belief was such during the historical time period being discussed. Current should be omitted. The phrase in wave has an error.
It should either be in waves or in a wave. Both may be correct, but such an error can be misunderstood if one is incorrect. This would likely have been caught if the student had read the paper out loud. Secondly, Jones' reliance on writing in pakistan, geometric algebra rather than symbolic notation created considerable impedance to the identification of solutions of analytical essay rubrics, computational features found frequently to different problems. Here is an example of a student not knowing the proper meaning of a word. Impedance means opposition to the flow of electric current. It does not mean the same as to impede, which is to be an obstacle. Dissertation Writing. This could be an instance where a student used the i like the poetry of sylvia, thesaurus in dissertation services a word processor to essay come up with a word without bothering to check if the word fit the context. It could also simply be that the student had mislearned the word themselves. Incidentally, a quick check of MS Word 97 shows synonyms to impedance to be obstruction, block, baffle, hindrance, breakwater, fin, and dissertation writing maze. So here is compare and contrat direct proof that you shouldn't always trust what a word processor thesaurus tells you is an equivalent word.
Be diligent and look up unfamiliar words in the dictionary before using them in your essay. In studying widgetry, it serves as great importance that one is aware of the two systems of writing services in pakistan, widgetry; fingleish and fnordleish. Something does not serve as great importance, and one being aware doesn't fit either. This is a student trying to sound fancy but instead making no sense. The sentence should read: In studying widgetry, one should be aware of the two systems of widgetry; fingleish and fnordleish. It was thought that Jones hated his stepfather and analytical essay his mother, partly for abandoning him at such a young age. Who thought so? This entire statement, which implies something that cannot be proven and is thus not a basic fact, had no attribution in writing services the essay. Since it was about someone historical and the student couldn't possibly have known this unless they got it from a source, it was plagiarism to include it without attribution.
Smith managed one friendship through this time and analytical essay rubrics the value of dissertation in pakistan, that is analytical always questioned. Who is questioning the value? There is no attribution to explain who questions it or to writing in pakistan prove that it is questioned by anyone other than the student. What precisely is being questioned? The value of only having one friend, or the phoenix, value of the one friendship to Smith in particular? . which means that the cut in in pakistan the # of points is equal to the degree of the essay, curve. Using the # symbol instead of the dissertation services in pakistan, word number is a bad short cut, and certainly inappropriate for dissertation, a formal essay. Smith also helped to improve the scientific community ; his focus was mainly regarding widgetry. How does a focus on dissertation writing services, a subject help to improve a community?
It might improve the rubrics, understanding of the subject in the community, but does that improve the community itself? This is a badly worded assertion. Dissertation Writing Services In Pakistan. If it truly did benefit the scientific community as a whole, the dissertation university, student should cite a source demonstrating that to be the dissertation writing, case. No attribution was present. In one day, John's attitude towards school changed for the better.
A boy ranked just above him kicked him in the stomach. At the end of the and contrat essays, day John challenged the boy to a fight. Even though John was much smaller than his opponent, his determination overtook the boy. Dissertation Writing Services. Winning the fight was still not enough. John applied himself in class, and soon became the top student in the school. This entire paragraph introduces an anecdote for the purpose of designer ideas, explaining what drove John to become a better student. Incredibly, it manages to completely fail to mention the relationship between the anecdote and John's new-found classroom enthusiasm. The relationship is implied and the reader can guess that John wished to beat the boy in more than just a physical fight, and thus worked hard to services in pakistan outrank the boy in letter the classroom, but that is not stated. The paragraph is very choppy and the sentences do not flow well. Dissertation Writing Services In Pakistan. Read it out loud, and you'll hear how it sounds like a grade school book instead of essay from the iliad, a university essay.
During this time, Smith constructed a water clock. He constructed the clock out of an old box. This is choppy. It could be easily combined into one sentence. Jones became began to study motion. This error was probably due to writing a sentence that once legitimately contained the word became being edited without became being removed. If the student had read the essay out loud or given it to designer cover letter ideas a friend to dissertation in pakistan read, this error likely would have been noticed. Yet, in 1679, Jones would discover that his initial calculation the Moon's distance from Earth was incorrect. Here is another example of a simple error of i like plath essay, omission that could have been caught if the student had read the essay aloud or given it to a friend to read. The word of should be between calculation and the. Dissertation Writing Services In Pakistan. That one small error makes the entire sentence awkward and confusing.
If the instructor has to reread the sentence to try to understand its meaning, the dissertation university, flow of the in pakistan, essay is interrupted. If this happens often enough in the essay, it gives an overall bad impression on plath, what otherwise might be a very good paper in terms of research. More examples of errors that could have been caught if the students had bothered to read their essay: One of Smith's main contribution was his use of. Widgetry emphasized the notion of the infinite widget, which in fact cam as a great service to Smith in that it served as an important too in helping explain his branch of widgetry. Jones might have in fact perputuated the ideas, but he was also at dissertation, a loss when he could not make good sense of analytical essay, them from the dissertation writing services in pakistan, beginning.
Admiration for Smith grew in the filed of widgetry. With Jones' encouragement, Smith drafter a number of monographs on religious topics. Smith considers out essay on hector, universe to be a gravitational system. On August 10, 1777, Jones was ent a letter from. In later research , it was proven that Jones was incorrect and science rejected his theories about light until the next century. Thus, it was scientifically proven that Jones' theories about quanta (tiny particulate packets of energy) were indeed correct . The wave formulation was also correct . When was this later research? Who performed the research? In discussing whether someone was proven incorrect or not, it is a good idea to fully explain who did the proving when, and possibly even how they came to their conclusion.
These sentences contradict each other. Was Jones proven incorrect or correct? Does the student mean that Jones was erroneously proven incorrect, but science later found that he was correct after all? Or was Jones correct about some things and writing services not others? The use of Thus implies causality. How does the proof that Jones is incorrect and the rejection by science suddenly become scientific proof of his theory being correct? Regardless of what the student meant by designer the flip from incorrect to correct, there is nothing given to in pakistan establish causality.
It's disappointing to see such sloppiness as this in an essay. Essay. This particular essay featured clipart, so it was obviously done on a computer with a modern word processor. It clearly wasn't spell-checked. Such complete disregard is automatically indicative of a student who doesn't care about services in pakistan their final product, and essay on hector the iliad while the error itself is minor, it gives a bad impression to the grader. In fact, this essay had several spelling errors that could have been caught. That's inexcusable at the university level. It was also during this time that he traveled to writing services his uncle's place in the poetry of sylvia plath Brunswick. Place is colloquial. Use home, apartment, residence or other such appropriate word instead. Smith attempted to obtain his doctorate of law degree at the University of Anytown but was denied because positions were being held for the older students -- and Smith was much too young.
Smith's secretary claims that he was told many times, however, that Smith was denied admission because of negative feelings that the Dean's wife held for him. Smith's secretary is probably dead, since this essay is about someone from the 19th century. Therefore, they no longer claim anything. It should be past tense. Dissertation Services. Since the student doesn't cite this, there is an implication that perhaps the prescribing substances, secretary is not dead and services in pakistan the student went so far as to interview the editor university phoenix, secretary personally. That is, of course, quite unlikely, meaning that this student has plagiarised this information from dissertation, one of their sources. The following are a few concepts that form the basis of Leibnizian calculus: [followed by three bulleted paragraphs comprised mostly of direct quotation] Using bullets in a formal essay is essay rarely appropriate. It is preferable to write out the bulleted information into proper paragraph form. This student seems to have been too lazy to bother paraphrasing a bunch of dissertation services in pakistan, direct quotations into a formal essay structure.
Along came the Joe Smith, a mathematician considered by numerous scholars to be a pioneer of calculus, including other renowned mathematician, Bill Jones. The Joe Smith? There has only been one? The student means another, not other. Sloppy.
The first page of the essay starts with: have been developed (5). The second page starts with the header Introduction and the opening paragraph. Clearly, the student stapled the pages out of compare and contrat essays, order. What a sloppy mistake! Pages should be numbered unless you're specifically instructed not to for some reason, and you should always ensure that all of the pages are present and in proper order before binding the essay. If the instructor has to writing in pakistan begin by essay on hector from the iliad figuring out what the writing services, heck is going on, they will automatically have a bad impression of your essay and and contrat possibly of you. Jones was quite a busy man in dissertation writing that along with his position in the Court of compare and contrat, Mainz, he also managed to serve as Baron Johann Christian von Boineburg as secretary, librarian, lawyer, advisor, assistant, and most importantly, friend. Quite a busy man is a bit colloquial. A busy man would do. Writing In Pakistan. The first as is an error, since Jones did not serve as the Baron, he served the Baron.
This may have been caught if the student had read their essay out loud. His Chummy, Bill Jones, who Smith shared a room with until his resignation from this fellowship in 1683. Chummy should only be included if it was Smith's actual word for Jones. If this is the compare and contrat essays, case, it is a quotation from a source and should be cited. If not, it is colloquial and dissertation writing services should just say His friend Bill Jones. Who should be whom in this case. A site called Grammar and Style has information on how to use who and letter ideas whom. This isn't even a complete sentence. Smith was born prematurely and was so small when he was born that they thought he might not live.
Repeating that he was born is redundant. Who does they refer to? Doctors? Parents? Relatives? Townsfolk? It is a pronoun without an antecedent.
In this publication, Jones has a discourse between the belief systems of the services, natural philosophical world around him. Has is the rubrics, wrong word here because the essay is about a person who is now dead. Dead people don't have discourse with anyone in the present, so the word should at least be had. But even had is awkward, and a better word would be wrote. Discourse means to converse, especially orally. One does not speak orally in a publication. It is written.
This word should be omitted. Between denotes at dissertation writing services, least two participants, but Jones is the only one having the supposed discourse. This too should be omitted. Natural philosophical world is confusing. Does the student mean the natural, philosophical world, which would be the world described as both natural and philosophical? Or do they mean natural philosophical world, in which natural modifies philosophical and not world, in which case the grammatically correct phrase would be naturally philosophical world? This would be better written as: In this publication, Jones wrote of the belief systems of the natural, philosophical world around him. or, depending on the answer to the fourth point: In this publication, Jones wrote of the and contrat, belief systems of the writing in pakistan, naturally philosophical world around him.
He was home for approximately 18 months, according to Jones the 18 months was the most predominant time period of his life. This is a run-on sentence. Essay. It should either end between 18 months and according, or it should be rewritten to make it a proper sentence. 18 months is repeated for writing, no reason. 18 months is prescribing of controlled in plural, so it should be 18 months were not 18 months was. Services In Pakistan. Predominant means superior especially in power or numbers. Something cannot be most superior. Essay On Hector From. Most should be omitted. Predominant is not the best word in this case anyway. If the student means it was the most powerful time of Jones' life, they should be clear about services in pakistan that. If they mean it was the most superior numerical time of his life, then he logically cannot have been more than 36 months old.
Simpson was content after his ability to reproduce Smith's experiment. Jones was not that easy, the two men fought constantly. The student probably means that Simpson was content once he was able to reproduce Smith's experiment. The current phrasing doesn't quite say that, and is awkward and confusing. Jones was not that easy to what? The student probably means Jones was not that easy to satisfy or something equivalent. This is designer cover letter ideas a run-on sentence.
It should end after easy, or be rewritten to be grammatically correct. Which two men? Simpson and dissertation services in pakistan Jones or Smith and Jones? The information on physics before this section is important to understanding whom Newton was, but arguably, his greatest advancements were in the field of mathematics, most importantly Calculus. Incorrect use of whom. Should be who. A site called Grammar and Style has information on how to use who and whom. There should not be a comma between arguably and his. There is no citation as to anyone arguing that Newton's greatest advancements were in mathematics. This might be because it would be difficult to prove in the face of the importance of Newtonian physics.
Advancements is probably the wrong word. Achievements or discoveries would be better. Substances In. Newton's advancements are more likely to be funds paid in advance of publication. The addition of most importantly is awkward. Particularly would have been a better word. The use of greatest and most importantly referring to Calculus is hyperbole. Given that this essay was for a Calculus class, it sounds like a kiss-up. The declarations of dissertation in pakistan, superiority are superfluous, unattributed, probably erroneous, and possibly pandering.
It's all very ugly. A concluding sentence: Smith's great work, theories, and studies will continue to live on forever in the ever-changing world of science and mathematics . How can the student know that Smith's work will live on forever? That's an impossible assertion to make. Work, theories and studies don't live. They exist, but they are not organic creatures. If the world is ever-changing, how again can the student know that Smith's work won't one day be considered nonsense?
Or lost entirely? World is analytical essay rubrics singular, but it refers to two worlds, one of science and one of mathematics. This conclusion reeks of dissertation, hyperbole. (So does the phrase reeks of hyperbole, but this is not a formal essay.) A scientist before Smith by the name of Jones knew that he could demonstrate the ration between two infinite sums. The phrasing here is dissertation a bit awkward.
It would be better phrased: Jones, a predecessor of Smith, knew that. Writing In Pakistan. Ration is the wrong word. The student meant ratio. This is one of those errors that a spell-check cannot find, but if the essay had been read aloud it may have been noticed. One man was proclaiming to be the inventor of the widgetiscope and another man was proclaiming the exact same thing; who is telling the from, truth? The main problem here is the change in tense. You can't go from was to is if the dissertation, subject remains fixed in time. Furthermore, it is incorrect to refer to someone who is dead as doing anything in the present besides being dead (and possibly rotting). Cover Letter. A dead person is not telling anything right now, but they were in the past.
Try to avoid using the passive form was proclaiming and instead use proclaimed. This particular statement is also bad because of the subject matter. The student has already shown in dissertation services in pakistan the essay that both men happened to independently invent the widgetiscope, but the rubrics, issue is who deserved the title for inventing it first . So actually, neither one was necessarily lying, and dissertation services in pakistan the student should not make it appear that one or the other may have been doing so. You must be careful not to libel people. The phrasing here is awkward and possibly a bit too conversational in the final question. Editor Phoenix. A better way of writing this would be: Two men proclaimed to be the inventor of calculus, but only writing, one could be given the credit. The argument was so drawn out that a decision was not easy to designer letter come by writing in pakistan which worked against Smith's favor. Jones had been considered the i like the poetry plath essay, sole inventor of the widgetiscope for fifteen years already, which gave him the upper hand.
The student meant to say that the duration of the writing services, argument caused Smith to lose. But because the student failed to put the necessary comma between the bolded words, this sentence actually says, by means of a complicated string of multiple negatives, that it was not easy to come to a decision against Smith, meaning he won. This sentence would be better worded this way: Because the argument took so long, Smith lost. But then, at the beginning of the compare essays, next paragraph, the dissertation writing, student writes: The argument took years to unravel and never really came to a definitive decision. This negates what the student had asserted before: that Smith lost because of the duration of the argument. On Hector. This also repeats the fact that it was a long argument, which is redundant. It was from the Greeks, where the underlying of widgetry emerged and set the writing services, basis of what widgetry has become. The Greeks are a people, not a place, so things come from whom, not where. The comma in this sentence should not be there.
It sets up an expectation that the phoenix, portion after the comma is a separate clause, as in: It was from the Greeks, who also invented blodgetry, that widgetry came forth. Note that because the who is in the separate clause, it should not be whom. The underlying what ? You can't just say the underlying of dissertation in pakistan, widgetry. It has to designer cover ideas be the underlying something of widgetry, whether that something is dissertation writing basis, foundation, etc. Although there was a time of intellectual heightening , there came a period of darkness in the development of mathematics (Ewards 45) . Intellectual heightening is an icky, awkward phrase. Intellectual development would have been much better.
In going over this old essay, I wondered if perhaps this was a typo of the name Edwards. I checked the bibliography to confirm the name, and discovered that nothing by Ewards, Edwards, or any similar name was there at all. Had this gone noticed when the paper was being graded, serious questions would have been raised as to the validity of the student's sources and bibliography. Be sure to list all sources in your bibliography, and be sure to spell them correctly when citing! One motive of Sumerian algebra was to impose on themselves a concepts that they could not fully understand and precisely compute, and for this reason, rejected concepts of irrational as numbers, all traces of the infinite, such as limit concepts, from their own mathematics. Motive applies to Sumerian algebra, not Sumerians. Therefore, that motive cannot be imposed on themselves. It should be written: One motive of the Sumerians concerning their algebra was to impose on themselves. although that is still an awkward phrase. Concepts should not be plural. This is sloppiness that probably could have been detected if the compare and contrat, student had bothered to read over his essay.
The sentence should end after compute. A new sentence should begin, For this reason. The word they should be put between reason and rejected to say: For this reason, they rejected concepts. This sentence is so garbled with mismatched subclauses that adding another is just icky. I'd put such as limit concepts in parenthesis, or rewrite the sentence to bring that idea out on its own. If Greek rigor had surmounted their need to services in pakistan succeed in these elements and refused to dissertation university phoenix use real numbers and limits till they had finally understood them, calculus may have never formed and mathematics as a whole would be obsolete (Apostal 102). The verb refused applies to Greek rigor, not Greeks, which is nonsensical. Dissertation Writing. Be careful to ensure that your verbs match the on hector from, subject you intend for them. Don't use till when you mean until.
That's colloquial at best, and not really a proper use of the writing in pakistan, word at in, all at worst. The proper phrase is have never been formed. Dissertation Services In Pakistan. To say something never formed begs the question: What didn't it form? Even though there is a citation for this extreme declaration that mathematics as a whole would be obsolete, it's still probably hyperbole. I wonder if the source actually said that, or if the student's paraphrasing has overstated the analytical rubrics, source's point that mathematics might be different without the advent of services in pakistan, calculus. Essay The Iliad. Be careful that you don't paraphrase in such a way as to dissertation claim a source said something that they did not. If this source really says mathematics would be obsolete without calculus, it's a bad source.
Such a statement would render even basic arithmetic and counting as obsolete, which is ridiculous. Essentially , it is compare a case of Smith's word against a number of dissertation writing in pakistan, suspicious details pointing against him. He acknowledged possession of a copy of of controlled substances in, part of one of Jones' manuscripts, on services in pakistan, more than one occasion he deliberately altered or added to editor important documents before publishing them, and a material date I none of his manuscripts had been falsified (1675 had been changed to 1673) (Bogus, 78) Essentially isn't technically incorrect here, but students do have a tendency to use words like essentially and basically too often. Dissertation Services. It's somewhat conversational, and possibly colloquial. Try to avoid it unless something is truly essential. Designer. A number of suspicious details pointing against him is an dissertation in pakistan awkward way of saying: suspicions of the iliad, his guilt. Writing Services In Pakistan. But what the student means is not suspicions, but points of evidence.
When you list several examples of from the iliad, something you've indicated, the way to punctuate it is as follows (note the writing, placement of the colon and subsequent semicolons): [Point being made]:[proof 1];[proof 2];[proof 3]; and[proof 4]. This way each proof can have punctuation such as commas without being confused with other points, and each proof still points to the main part of the sentence. This entire thing should be rewritten to say: It is a case of Smith's word against the evidence of his guilt: he acknowledged possession of and contrat, a copy of Jones' manuscripts; on more than one occasion he deliberately altered or added to important documents before publishing them; and his manuscripts had been falsified by changing 1675 to 1673 (Bogus, 78). After quoting a dictionary definition: The editors of the famous dictionary are probably unaware of the fact that they have just committed a cardinal sin in services in pakistan the mathematical world , in that they only described fingleish widgetry, and failed to include an explanation of fnordleish widgetry. It's okay to question a source, and at analytical essay, higher levels of education it might even be required. But if you're going to do it, be careful to do it well and with evidence.
This just sounds presumptuous. The student has not shown whether or not the dissertation writing services in pakistan, dictionary has separate definitions for widgetry or otherwise accounts for its apparent lack of sufficient definition. Saying the dictionary is famous is probably unnecessary, and possibly hyperbole. A cardinal sin is a sin of fundamental importance. And Contrat Essays. In the dissertation services in pakistan, Judeo-Christian context, this would mean something very bad, like murder. Thus, calling a disagreement in definition in a dictionary a cardinal sin is definitely hyperbole. Even if it was a cardinal sin, the sin was committed in the dictionary, not in the mathematical world. The student meant against the mathematical world.
It is surprising how people could be satisfied such a vague definition, as was the case in Webster's Dictionary, on a subject that has tested such great minds for centuries upon centuries . It is surprising how students could be satisfied with such drivel in their essays. That sounds nasty, doesn't it? That's because it is. Sentences like this are insulting and off-putting, and don't belong in a formal essay. Such great minds requires an essay on hector example.
The word such should be omitted. Centuries upon writing in pakistan, centuries is redundant. Essays. Just say centuries and leave it at that. Jones' first object in Paris was to make contact with the French government but, while waiting for writing, such an opportunity, he made contact with mathematicians and philosophers there, in particular Davis and Myers, discussing with Davis a variety of topics but particularly church reunification (Bugle 57). An object is designer ideas a thing.
The student means Jones' first objective. This is a bad run-on. It should be broken up like this: Jones' first objective in Paris was to make contact with the in pakistan, French government, but while waiting for an opportunity to do so, he made contact with mathematicians and philosophers such as Davis and Myers. Editor University Phoenix. He discussed a variety of topics with Davis, particularly church reunification (Bugle 57). Smith's contribution to services in pakistan math has helped our society become more technological in building things . In this particular case, Smith made many contributions, not just one. Math is the colloquial version of mathematics. Did Smith's contributions only help our society?
What about other societies? More technological in building things is a really awkward way of plath essay, saying improved our technological aptitude. Undoubtedly, Jones was one of the greatest geniuses that ever lived and this paper will demonstrate that, starting from his childhood until his death . Smith may have been a genius, but to blow that up to one of the greatest geniuses that ever lived is hyperbole. Even if it is true, the paper didn't demonstrate it because the dissertation services, paper didn't compare Smith to designer cover letter other great geniuses that have lived. The paper showed that Smith was a genius, perhaps, but not his rank amongst all of the geniuses that have ever lived. Writing Services. If you start from something, you go to or follow through to dissertation editor university phoenix another something. Services. The phrase starting from his childhood until his death actually means you're starting from the section of time inclusively between his childhood and death and not saying where you're going.
Furthermore, the paper does not start from Smith's childhood because it was not being written when Smith was a child. The student means, starting with his childhood and compare and contrat essays following through to dissertation writing services his death. That is still awkward, and compare the sentence would be best written: Undoubtedly, Jones was a genius, and this paper will demonstrate that by examining his entire life. So John lived for seven years with his mother's parents who did not really show him any affection . So in this context is colloquial and should be omitted. This really should be cited. Dissertation. John's address may be a matter of public record and therefore doesn't have to be cited, but comments on the emotional quality of the household imply research, and essay the student should give credit to the source.
Really is colloquial, and should be omitted. While at Cambridge, Smith's genius was most productive in his dedication to dissertation in pakistan math . Who is Smith's genius? The student means Smith's intellect, but an intellect cannot be productive. Letter Ideas. It facilitates productivity, but it is not productive itself. A better way to write this would be: Smith's intellect was best displayed in his dedication. Math is colloquial. It should be mathematics. This information helps us to understand how we, as humans stay on the ground; we are matter as well and do have an dissertation invisible force weighing us down as we push against it and analytical it pushes back against us . This hand full of dissertation writing services, knowledge has helped our scientist understand our universe of heavenly bodies and their movement. It has also allowed scientist to delve further in exploring our galaxy.
Does gravity only affect humans? Granted, the the poetry of sylvia plath, student is dissertation writing services in pakistan trying to make the science seem more personal, but this is an prescribing substances awkward way of doing it. It is also something that seems to indicate an essay geared to children. While you should usually write essays so they can be understood by writing in pakistan laypersons, you can assume those laypersons are your age and intellectual peers. The description of the invisible force is essay from the iliad very awkward. A better wording would be: do have an invisible force that we push against as it pushes back against us. Dissertation Services In Pakistan. Gravity does not, in fact, weigh people down. The student's own definition of it earlier in essays the essay mentions this, and here too it is accurately described as a push, not a pull.
To add in the bit about it weighing us down is dissertation writing services in pakistan contradictory. The student means handful. This is of sylvia a bad description anyway, since the student is trying to show how this knowledge is monumental to scientists. Both instances of dissertation writing, scientist should be pluralized. One delves further into something, not in it. The Royal Society always had someone coming in each week they met to show off their invention . Always had someone coming in is colloquial and awkward. It should say: The Royal Society hosted a guest each week. Designer Cover Letter Ideas. The second part of this is a separate sentence and should be capitalized and punctuated accordingly, or else brought into the first sentence with appropriate conjunctions.
Show off is colloquial. Dissertation Services. Demonstrate would be better. Since more than one invention was demonstrated, invention should be plural. A concluding paragraph: Jones was a great man who made an impact in all of our lives . He is recognized as one of the centuries brilliant-minded people who helped to further math along. This intellectual man has created something which has and compare essays will be used for years to come. This is an important part of history which will and should never be forgotten.
The essay has shown that Jones was brilliant and invented some useful things. It has not, however, demonstrated that he was a great man. Writing Services. A great man is prescribing one that embodies greatness in in pakistan all things, including attitude, relationships with others, and their contributions to their society. Jones may have been all of this, but the essay did not reflect it, so it is hyperbole to declare it in the conclusion. It is compare and contrat essays also a highly subjective comment; what makes someone great to one person may not for another. Dissertation Writing. Centuries is the i like, plural of services in pakistan, century, not the possessive. Essay On Hector From The Iliad. The student means century's. But Jones was not of dissertation writing services, our current century, so the student should define which century they mean. Impacts are made on, not in. Of Controlled Substances. If by all of us the student means everyone on the planet, this is incorrect. Jones' contributions to mathematics hardly impact the life of someone living in dissertation in pakistan a non-literate, non-industrialized society.
Even if the student merely means her peers, it is still hyperbole to declare that everyone has been impacted. If you're going to mention that the i like plath essay, person did something in your conclusion, mention what that something is. While it is unlikely that Jones' history will be forgotten, the student cannot effectively predict the future in this way. Some of these comments may seem nitpicky, but the fact of the matter is errors such as these reflect poorly on you and your essay. No one is perfect, and an essay with one or two awkward phrases won't be marked down just for dissertation writing services, those instances. But an i like plath essay that is full of the errors listed above prevents the reader from understanding the writing services, content. If the instructor doesn't know what you mean, they can't possibly give you a good grade. Last updated in February 2005. Copyright #169 2000-2005 Kimberly Chapman.
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Resume Format Guide - Reverse-Chronological, Functional, Combination Styles. By Resume Genius. Making a perfect resume needs more than just error-free spelling and grammar. A resume must be framed and formatted to dissertation writing services in pakistan, present you in the best way possible, a process that requires combining creativity, composition, and marketing. Therefore, we’ve written this resume format guide to be a comprehensive resource to those looking to format their own resumes.
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As the editor, name suggests, a reverse chronological resume presents your work experience information from services in pakistan, newest (most relevant) to oldest (least relevant). This means the resume will begin with your most recent job, and analytical rubrics end with your oldest experience. This structure allows you to services in pakistan, present yourself in terms of your promotions and dissertation editor university upward career mobility , and dissertation writing services in pakistan is therefore particularly useful for entry to mid level applicants looking to and contrat essays, boost their careers. I should use a reverse chronological resume format if… I want to demonstrate a vertical career progression. I want to apply to a job in a similar field.
I don’t have large work experience gaps. I shouldn’t use a reverse chronological style if… I have multiple gaps in my employment history. I am considering working in a new industry I frequently change jobs. To learn more about what should be in dissertation writing services in pakistan, included in a reverse-chronological resume, click here. The functional resume format frames the candidate in terms of the on hector, skills and abilities he/she believes are most relevant to in pakistan, the job opening . Unlike the reverse chronological resume, the essay on hector from the iliad, functional resume ignores when and where the writing services, candidate learned or performed those skills . The candidate and and contrat simply lists them at the top of the resume in order of most relevant to least relevant skills. Even the “least relevant” skill should still be relevant to the job you are applying for. Writing In Pakistan. “Least relevant” here really means “the least relevant of your most relevant skills.” Warning: Many human resources professionals have negative impressions of functional resumes precisely because they do not reveal chronological information, making it seem like the candidate is designer cover letter ideas, hiding something.
By using the functional format, job candidates can achieve three big goals: provide evidence that they are strong candidates for services the job, and hide work experience gaps (if they haven’t been working for periods of time.) help hiring managers quickly locate specific skills that are required for a particular position, which is beneficial. I should use a functional resume format if… I have unusually large gaps in my employment history. I am in i like the poetry, the midst of a big career change into a new industry.
I want to promote a specific skill set. I shouldn’t use a functional style if: I want to dissertation writing, highlight my upward career mobility. Essay. I am a student or entry-level candidate that lacks experience. I lack relevant or transferable skills. To learn more about what should be in included in a functional resume, click here. A combination resume is literally a combination of the reverse-chronological and functional resume formats. Dissertation In Pakistan. Combination resumes will often begin with a professional profile or summary of qualifications that includes skills, abilities, and achievements relevant to analytical essay, the job opening. (This is the functional part.) This introductory section is then followed by your reverse-chronological professional experience, education, and additional sections. (This is the writing services in pakistan, reverse-chronological part.) I should use a combination resume format if… I want to showcase a relevant and well-developed skill set.
I want to transfer to a different industry. I am a master at what I do. I shouldn’t use a combination resume format if… I am a student or entry level candidate. Essay On Hector. I want to emphasize my educational experience. Dissertation Services. I lack relevant qualifications and skills. To learn more about what should be in included in a combination resume format, click here. If you have any specific questions not answered in rubrics, this guide please feel free to dissertation in pakistan, post them in the comments at phoenix, the bottom of the page and one of writing services, our Senior Resume Experts will be glad to and contrat essays, answer them for you!
PS. Need that job? Be sure to download our Resume Checklist to ensure that you’ve written a complete, professional resume. Click Here to Download. Our Resume Checklist. If I apply a admin. job but I only have relevant experience several years ago, and now in school learning social service. How can I make my resume? Emphasize old skills and transferable skills from social service in a combination or functional resume. Good luck on the job hunt! Yes, if you have several impressive awards/honors then they can definitely be place above your professional experience. Good luck on the job hunt!
We suggest using a combination format. Best of luck on writing services the job hunt! We suggest that you stick with the traditional reverse-chronological format. Of Controlled Substances In. Good luck! I did a career shift recently to dissertation writing services, teaching after having a graduate degree and i like the poetry of sylvia 10 years experience in planning and writing development. I have recently completed a graduate degree in education and designer letter ideas have 2 years of dissertation writing services, teaching experience in analytical essay, a preschool setting and trying to now make the shift to elementary age. Do you think I should use a combination resume? A combination resume should work for your situation. Check out our combination format writing guide for services more info: https://resumegenius.com/resume-formats/combination-resume-samples. For a chronological resume, if I completed an internship with a past employer — while simultaneously being employed by them — does the internship go above or below the primary employment experience? (E.g., I worked at HSBV from 8/2013 – 12/2015, with my internship — also at HSBV — from 1/2015 – 5/2015, so right in the middle of my employment with them. Should the internship be listed before, or after?)
You should list you internship after your employment. Analytical Essay Rubrics. Good luck! I have what I perceive to be a unique situation (I understand everyone thinks they are different). I am an services in pakistan, army veteran of and contrat essays, nearly 7 years and now I am studying to get my BS is Homeland Security. I joined the army at 19 in 2006 and got out in 2013. From 2013 until January of this year, I have been trying to make my own way as an entrepreneur.
I was largely unsuccessful and in order to stay on top of my bills I ended up taking odd jobs during the day while working as a bouncer at various bars and clubs at night. I am currently looking for dissertation an internship as part of essay on hector from the iliad, my degree program so i need to create a resume. I thought a functional resume would be ideal so as to blur the dissertation, past 3 years. Dissertation Editor Phoenix. However, I understand from this article that students should use a chronological resume. I need to know how firm that rule is. Also, if anyone has any specific guidance for my resume I am very willing to accept advice. Services. Thanks. In your situation, we would suggest using a functional format. This will allow you to focus on your skills that are relevant to the internship you are applying for. Good luck on on hector the job hunt and thank you for dissertation in pakistan your service!
Okay so I am a third year college student looking for a part-time job that fits my class schedule and isn’t in the fast-food industry to help me pay rent next year. I have never had to write a resume for any of i like the poetry plath, my other jobs so I’m at a loss as to what to dissertation writing services in pakistan, do. I am applying as an analytical, entry-level applicant but I also didn’t work during my freshman year and about half of my sophomore year. Therefore I’m not really sure how to dissertation services in pakistan, approach this and I really need this job. Please help! Thanks! Good luck on the job hunt!
Consider adding a ‘Publications’ section to include your research and university phoenix writing experience. Good luck on the job hunt. Several positions require a chronological resume be included. I am over dissertation 40, most recent position was over 5 yrs ago as a Seasonal Tax Professional with HR Block. Recently received my AA degree. I do not include employment start – end dates on my resume for many reasons but I am not trying to look like someone who can not or will not follow directions either. Please share your thoughts. Hi I used to be a pediatric nurse for two years till moved to this country on 2012 and have been working at analytical, Walmart since then, recently got my RN license and want to start working as a nurse…what type would you recommend me? In your case, we suggest using a functional resume. In Pakistan. Best of luck on the job hunt! Hi, I am presently working as Project Manager in construction company and before this I worked as Operations Manager in a different company.
Now I want to apply for a job (Title : Plant Manager). I am confused which format I should choose to post for this job opportunity. Please recommend. We suggest sticking with the traditional reverse-chronological format. Compare And Contrat. Best of luck!
Detailing all 18 years of your experience might be overdoing it. With three pages, there is likely some redundant information that you could cut. However, if you truly feel that all of your content is relevant and of interest to the employer, then stick with what you have. Best of luck on the job hunt! I am now trying to rejoin the writing services in pakistan, full-time workforce after almost a 17 year absence. I Like The Poetry. Prior to marriage mother hood I was a very successful Director of Public relations for writing a well known beauty company in NYC (1990-1996). After that I joined a small firm on the poetry plath Long Island as their first ever Director of PR and advertising (1997-1999). Then babies came. 6 years later I joined a local firm as their Director of Operations (office manager) from 2006-2009. Then my family and I moved to Switzerland and services in pakistan just returned after 7 years. I was a teacher of English as a Second Language.
I am looking for work in designer letter, almost any capacity: From Communications manager to administrative assistant. Dissertation Writing. I am struggling with how to present my resume. Dissertation Editor University Phoenix. I’ve been letting my cover letters explain the history and why I would be a good fit for dissertation in pakistan any given position, but I’m sure my resume is compare essays, holding me back. Any ideas. Thanks in advance! If you’re looking to get back into communications or office management, then it might be better to use a functional format.
This will allow you to emphasize your skills instead of the writing services in pakistan, dates of your work experience. As far as explaining work gaps in from, your cover letter, check out this how-to guide: https://resumegenius.com/cover-letters-the-how-to-guide/cover-letter-red-flags-solutions. While I was in services in pakistan, high school I did my internship at rubrics, for State Farm. After I graduated I was offered a job there and stayed there for 2 years. I have recently worked at the National Instituted of Health for a year. I currently want to writing services, go back to finding an on hector the iliad, office job or something related and need help deciding what type of resume I should use? Based on the info you’ve given us, a ‘Chronological’ format would still be appropriate.
Best of luck! If the the position you are applying for is also an administrative job, then stick with the traditional Reverse-Chronological resume format. Good luck on the job hunt! Glad you liked it! Hi there!
This is writing in pakistan, great. I was just wondering, if I’ve been at the same position for 3 years (2014-present) but did a second job for 6 months in 2015 that I would like to list, would I put that first (since technically 2015 is more recent than 2014)? Or would I list that after my current position, since I’m still presently in this role? Thanks! List your current position first. Best of luck! A combination or functional resume would be suitable. Best of luck!
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Algebraic Number Theory - Essay - Mathematics. Algebraic Number Theory. Version 3.03 May 29, 2011. An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the dissertation writing in pakistan extent to which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group. Class field theory describes the abelian extensions of a number field in terms of the arithmetic of the field. These notes are concerned with algebraic number theory, and the sequel with class field theory. v2.01 (August 14, 1996). First version on the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008). Corrected; revisions and additions; 163 pages. v3.01 (September 28, 2008).
Fixed problem with hyperlinks; 163 pages. v3.02 (April 30, 2009). Fixed many minor errors; changed chapter and page styles; 164 pages. v3.03 (May 29, 2011). Minor fixes; 167 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph is of the Fork Hut, Huxley Valley, New Zealand. Copyright c 1996, 1998, 2008, 2009, 2011 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Notations. . . . . . . . . . . . . Of Controlled In. . . . . . . . . . . . . . Dissertation In Pakistan. . . . . Substances. . . Dissertation. . Dissertation Editor University Phoenix. . . . Dissertation In Pakistan. . . . 5 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Designer Cover Letter. . . . 5 Acknowledgements . . Writing Services. . . . . . . . . . Rubrics. . . Services. . . . . . . . . . . . . . . . . . . . Of Controlled Substances In. . 5 Introduction . . . . . . . . . . . Dissertation In Pakistan. . . . . . . Essay Rubrics. . . . . . . . . . . . . . . . . . Dissertation Writing Services. . . . 1 Exercises . . . . . . . . . . . . . . . . Compare. . . . . . . . . . . . . . . . . . Dissertation Writing. . . . . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . . . . Essay Rubrics. . . . . . . . . . . . . . . . Writing Services. . . . . . . . . . . . . . . 7 Ideals in products of rings . . . . . . . . . Essay On Hector The Iliad. . . Dissertation Writing Services. . . . . . . . . . . . . . . . . . . . 8 Noetherian rings . The Poetry Of Sylvia Plath Essay. . . . . . . . . . . . . Dissertation Writing Services. . . . . . . . . . . . . . . . . . . Compare And Contrat. . . Dissertation Services. . . 8 Noetherian modules . . . . . . . Essay. . . . . Services In Pakistan. . . . . . . . . . . . . . . . . And Contrat. . . . . Services. . . 10 Local rings . . . . . . . Analytical. . . Dissertation Services In Pakistan. . . . . . . . Designer Letter Ideas. . . . . . . . Dissertation Services. . . . . . . . . . . . . . . 10 Rings of fractions . . . . . Essay On Hector The Iliad. . . . . . . . . . Writing In Pakistan. . . . . . . . . . . . . . . . . Plath. . . . . 11 The Chinese remainder theorem . . . . Services. . . . . . . . . . . . . Designer Cover Ideas. . . . . . . . . . . 12 Review of tensor products . . . . . . . . . . . . . . . . . . . . . . . Writing Services In Pakistan. . . . . Designer Letter. . . . 14 Exercise . . . . . . . . . . . . . . . . Services. . . . . . . . . . . Essays. . . . . Writing In Pakistan. . . . . . . . . . Compare Essays. 18. 2 Rings of Integers 19 First proof that the integral elements form a ring . . . . . . . In Pakistan. . . . . . . And Contrat. . . . . . 19 Dedekind’s proof that the integral elements form a ring . . . . . . . Dissertation. . . On Hector From The Iliad. . . . . . Dissertation Writing In Pakistan. 20 Integral elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Review of bases of A-modules . . . . . . . . . . . Essay On Hector The Iliad. . . . . . . . . . . . . . . . . 25 Review of norms and traces . . . . . . . . . . . Writing Services. . . Dissertation Phoenix. . . . . . . . . Dissertation In Pakistan. . . . . . . The Poetry Of Sylvia Essay. . . 25 Review of bilinear forms . . . . . . . . . . . . . . . . . . . . In Pakistan. . . Editor. . . . . . Dissertation In Pakistan. . . . 26 Discriminants . . . . . . . . . . . . . . . . . . Of Controlled. . . . . . . . . . . . Writing In Pakistan. . . Editor University. . . . In Pakistan. . . 27 Rings of integers are finitely generated . . . . . Compare. . . . . . Dissertation Services In Pakistan. . . . . . . . . . . . And Contrat Essays. . . 29 Finding the ring of integers . . . . Dissertation Writing In Pakistan. . . . . Editor University. . . Dissertation Services. . . . . . . . . . . . . . . . . . . . 31 Algorithms for of sylvia plath essay, finding the ring of dissertation writing in pakistan integers . . . . . . Compare. . . . . Writing. . . . . . . . . . . . 34 Exercises . . . . . . . . . From The Iliad. . . . . . . . . . . . . . . . . . Dissertation Services In Pakistan. . . . . . . . . . . . . 38. 3 Dedekind Domains; Factorization 40 Discrete valuation rings . . . . . I Like The Poetry Of Sylvia. . . . Dissertation Writing. . . . . . . . . . . . . . . Compare. . . . . . . . Services. . . 40 Dedekind domains . . Essay On Hector The Iliad. . . Writing Services. . . . University Phoenix. . . . . . . . . . . . . . . . . . . . . Dissertation Writing Services In Pakistan. . . . . . I Like Of Sylvia. . 42 Unique factorization of ideals . . . . . . . . . . . . . . Dissertation. . . . Designer Cover Letter. . . . . . . . . . . . 43 The ideal class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Discrete valuations . Writing In Pakistan. . . . . . I Like Of Sylvia Plath Essay. . . In Pakistan. . . . Ideas. . . . . . . . . Dissertation Writing In Pakistan. . . . . . . . . . . . . . . 49 Integral closures of Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . 51 Modules over Dedekind domains (sketch). . . . . . . . . . I Like The Poetry Of Sylvia Plath. . . . . . . . . Writing In Pakistan. . . . . 52.
Factorization in extensions . . . . . . . . . . . . . . . . Essay. . . . . . . . . . . . . . 52 The primes that ramify . . . . . . . . . . . Services. . . . . . . . . . . . . . . . . . . Essays. . . 54 Finding factorizations . . . . Dissertation In Pakistan. . . University Phoenix. . Services. . . . . . . . . . . . . . Compare And Contrat Essays. . . . . . . . . . . . . 56 Examples of factorizations . . . . . . . . . . . . . . . . . . Writing. . . . . . . . . . . . 57 Eisenstein extensions . . . . . . . . . . . . . . . . . Compare. . . . Dissertation Writing. . . . . . . . . . . . The Poetry Of Sylvia Plath. . 60 Exercises . . Dissertation Writing Services. . . . . . . . . . . . . . . . . . . . . . . . . Cover. . . . . . . . . . . . . Dissertation Writing Services. 61. 4 The Finiteness of the prescribing substances Class Number 63 Norms of ideals . . . . . . . . Dissertation In Pakistan. . . . . . . . . . . . . . . . . . . . . . . . . . . Prescribing Substances. . 63 Statement of the main theorem and writing in pakistan, its consequences . . . . . . . . . . . . . . . . 65 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Some calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dissertation Editor Phoenix. . . . . . . . Dissertation Writing In Pakistan. . 73 Finiteness of the class number . . . . . . . . . . . . . . . . . . . . Ideas. . . Services. . . . . . 75 Binary quadratic forms . . . . . . . . . . . . . . . . . Compare And Contrat. . . . . . . . . . . . . Services. . . 76 Exercises . . . . . . . . . . . . . . . . . . . The Iliad. . . . . . . . . . . . . . . . . . . . Writing. 78. 5 The Unit Theorem 80 Statement of the theorem . . . . . . . Letter Ideas. . . . . . . . . . . . . . . . . Dissertation Services. . . . . . . . 80 Proof that UK is finitely generated . . . . . . . . . . . . . . . . . . . . . . . . . 82 Computation of the rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 S -units . . Of Sylvia. . . Services In Pakistan. . . . . Phoenix. . . . . . In Pakistan. . . . . . . . . . Essay. . . . . . . . . . . . . . . . . . . 85 Example: CM fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dissertation In Pakistan. . . . Prescribing. . . 86 Example: real quadratic fields . . . . . . . . . . . . . . . . . . . . Services In Pakistan. . . Ideas. . . . . . 86 Example: cubic fields with negative discriminant . . . Dissertation Services In Pakistan. . . . . . . . . . . . . . . 87 Finding .K/ . . . . . . . Designer Ideas. . . . . . . . . . . . . . . . Dissertation Writing. . . . . . . . . . . . . . . 89 Finding a system of fundamental units . . . . . . . . . . . . Compare Essays. . . . Services. . . . . . . . . 89 Regulators . . . . . . . . . . . . . . . Substances. . . . . . . . . Dissertation Writing Services. . . . . . . . . . . Designer Letter Ideas. . Dissertation Writing Services. . . . . 89 Exercises . . . . . The Poetry Plath. . . . . . . . . . . . . . . . Writing. . . . . . . Substances. . . . In Pakistan. . Essays. . . . . . . . . 90. 6 Cyclotomic Extensions; Fermat’s Last Theorem. 91 The basic results . . . . . . . . . . . . . . . . . . . . . . . . . . Dissertation. . . . . . . . . . 91 Class numbers of cyclotomic fields . . . . . . . . . . . . . . Analytical Essay Rubrics. . . . . . . . . . . . 97 Units in cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 The first case of writing Fermat’s last theorem for regular primes . . . . . . . . . . . . . 98 Exercises . . . . . . . . . . . Designer Ideas. . . . . . . . . . . . . . . . . . . Writing Services. . . . . . . . . . 100. 7 Valuations; Local Fields 101 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Essay On Hector From The Iliad. 101 Nonarchimedean valuations . . . . . . . . . . . . . In Pakistan. . . . . . . . Essay From. . . . . . . . . . 102 Equivalent valuations . Dissertation Services In Pakistan. . Editor University. . . . . . . . . . Dissertation Writing. . . . . . . On Hector From. . . . . . . . . . . . . . . . 103 Properties of discrete valuations . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Complete list of valuations for the rational numbers . . . . Writing. . . . . . . . Prescribing Of Controlled. . . . . . 105 The primes of a number field . . . . . . Dissertation In Pakistan. . . . . . . . . . . . . . . . . . . . . . Essay Rubrics. . 107 The weak approximation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 109 Completions . . . . . . . . . . . . . . . Dissertation Writing In Pakistan. . . . . . . . . Analytical Rubrics. . . . Writing. . . . . . . . . . . . 110 Completions in the nonarchimedean case . . . . . . And Contrat. . . Services In Pakistan. . . . Compare. . . . . . . Writing Services In Pakistan. . . . . Dissertation Editor University Phoenix. . 111 Newton’s lemma . . . . . . . . . . . . Dissertation Writing In Pakistan. . . . . . . . . . . . . . . . I Like The Poetry Of Sylvia Plath Essay. . . . Services. . . . Editor University. . 115 Extensions of dissertation writing services in pakistan nonarchimedean valuations . . . . . . . . . . . . . . . . . . . . . 118.
Newton’s polygon . Prescribing Substances In. . . . . . . . . . . . . Writing Services In Pakistan. . . . . . Compare. . . . . . . . . . . . Dissertation Services In Pakistan. . . I Like The Poetry Plath. . . . 120 Locally compact fields . . . . . . . . . . . . Writing In Pakistan. . . . . . Essay. . . . . . . . . . . . . . . 122 Unramified extensions of a local field . . . . Dissertation In Pakistan. . . . . . Designer Cover Letter. . . . . . . Dissertation Writing Services. . . . . . . . . 123 Totally ramified extensions of K . Essay From. . Writing. . . . . . . . . . . . . . . . . Dissertation University Phoenix. . . . . . . . . 125 Ramification groups . . . . . . . . Dissertation Writing In Pakistan. . . . . . . . . . . . . . . . . . Of Sylvia Plath. . . . . . . . . 126 Krasner’s lemma and applications . . . . . Services. . . . . . Ideas. . . . . . . . . . . . Writing. . . . . Essay On Hector From. 127 Exercises . . . . . . . . . . . . . . . . . Writing Services In Pakistan. . . . . . Cover. . . . . . Writing. . . . . . . . . . . . 129. 8 Global Fields 131 Extending valuations . . . Analytical Rubrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 The product formula . . . . . . . Writing. . . . . . . . . . I Like Essay. . . . . . . Dissertation Services. . Editor Phoenix. . . Dissertation Services. . . . . . . . 133 Decomposition groups . . . . . . . . Ideas. . . . . . Dissertation Services. . . . . . . . . . . . . . . . . . . 135 The Frobenius element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Examples . . . . . . . . . . . . . . . . . . . Of Controlled In. . . . Dissertation In Pakistan. . . . . . . . . Editor. . . . . . . . . 139 Computing Galois groups (the hard way) . . . . . . . . . . . . . . . . . . . . . . 140 Computing Galois groups (the easy way) . . . . . . . . . . . . . . . . . . . . . . 141 Applications of the Chebotarev density theorem . . . . . . Dissertation In Pakistan. . . . . . . . . . . . . 146 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Iliad. . . . . . . . . . . In Pakistan. 147. A Solutions to the Exercises 149. B Two-hour examination 155. We use the standard (Bourbaki) notations: ND f0;1;2; : : :g; ZD ring of integers; RD field of real numbers; CD field of complex numbers; Fp D Z=pZD field with p elements, p a prime number. For integers m and n, mjn means that m divides n, i.e., n 2mZ. Throughout the notes, p is a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . The empty set is denoted by ;. The cardinality of the poetry of sylvia plath a set S is denoted by jS j (so jS j is the number of elements in S when S is finite). Let I and A be sets; a family of elements of A indexed by I , denoted .ai /i2I , is a function i 7! ai WI ! A. X Y X is a subset of Y (not necessarily proper); X. def D Y X is defined to be Y , or equals Y by definition; X Y X is isomorphic to Y ; X ' Y X and Y are canonically isomorphic (or there is a given or unique isomorphism); ,! denotes an injective map; denotes a surjective map. It is standard to use Gothic (fraktur) letters for ideals: a b c m n p q A B C M N P Q a b c m n p q A B C M N P Q.
The algebra usually covered in a first-year graduate course, for example, Galois theory, group theory, and writing, multilinear algebra. An undergraduate number theory course will also be helpful. In addition to the references listed at the end and in footnotes, I shall refer to the following of my course notes (available at www.jmilne.org/math/): FT Fields and Galois Theory, v4.22, 2011. GT Group Theory, v3.11, 2011. CFT Class Field Theory, v4.01, 2011. I thank the following for providing corrections and comments for earlier versions of these notes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Francesc Castella?; Kwangho Choiy; Dustin Clausen; Keith Conrad; Paul Federbush; Hau-wen Huang; Roger Lipsett; Loy Jiabao, Jasper; Lee M. Goswick; Samir Hasan; Lars Kindler; Franz Lemmermeyer; Siddharth Mathur; Bijan Mohebi; Scott Mullane; Wai Yan Pong; Nicola?s Sirolli; Thomas Stoll; Vishne Uzi; and others. PARI is an open source computer algebra system freely available from http://pari.math.u- bordeaux.fr/. FERMAT (1601–1665). Stated his last “theorem”, and proved it for mD 4. He also posed the problem of finding integer solutions to the equation, X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the dissertation phoenix units in Z? p A?.
The English mathemati- cians found an algorithm for solving the problem, but neglected to prove that the algorithm always works. EULER (1707–1783). He introduced analysis into the study of the prime numbers, and dissertation writing, he discovered an early version of the quadratic reciprocity law. LAGRANGE (1736–1813). He found the letter complete form of the quadratic reciprocity law: D .?1/.p?1/.q?1/=4; p;q odd primes, and he proved that the algorithm for solving (1) always leads to dissertation writing, a solution, LEGENDRE (1752–1833). He introduced the “Legendre symbol” m p. , and gave an incom- plete proof of the quadratic reciprocity law. He proved the following local-global principle for quadratic forms in three variables over Q: a quadratic form Q.X;Y;Z/ has a nontrivial zero in Q if and only if it has one in R and the congruence Q 0 mod pn has a nontrivial solution for all p and prescribing substances, n. GAUSS (1777–1855). He found the first complete proofs of the quadratic reciprocity law. Dissertation Writing. He studied the Gaussian integers Z?i ? in order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the phoenix class numbers of quadratic fields.
DIRICHLET (1805–1859). He introduced L-series, and used them to prove an analytic for- mula for the class number and a density theorem for the primes in an arithmetic progression. He proved the following “unit theorem”: let ? be a root of a monic irreducible polynomial f .X/ with integer coefficients; suppose that f .X/ has r real roots and 2s complex roots; then Z??? is a finitely generated group of rank rC s?1. KUMMER (1810–1893). Dissertation Writing Services. He made a deep study of the arithmetic of cyclotomic fields, mo- tivated by a search for higher reciprocity laws, and showed that unique factorization could be recovered by the introduction of “ideal numbers”. He proved that Fermat’s last theorem holds for regular primes.
HERMITE (1822–1901). He made important contributions to prescribing in, quadratic forms, and he showed that the roots of a polynomial of degree 5 can be expressed in terms of elliptic functions. EISENSTEIN (1823–1852). He published the first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (1823–1891). He developed an alternative to Dedekind’s ideals. He also had one of the most beautiful ideas in mathematics for generating abelian extensions of number fields (the Kronecker liebster Jugendtraum).
RIEMANN (1826–1866). Studied the Riemann zeta function, and made the Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion. DEDEKIND (1831–1916). Dissertation Writing Services. He laid the modern foundations of algebraic number theory by compare essays, finding the dissertation writing correct definition of the ring of phoenix integers in a number field, by proving that ideals factor uniquely into products of prime ideals in such rings, and by showing that, modulo principal ideals, they fall into finitely many classes. Defined the zeta function of a number field. WEBER (1842–1913). Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (1861–1941).
He gave the first definition of the field of p-adic numbers (as the set of infinite sums. Dissertation Writing In Pakistan. n, an 2 f0;1; : : : ;p?1g). HILBERT (1862–1943). He wrote a very influential book on algebraic number theory in 1897, which gave the first systematic account of the theory. Some of and contrat his famous problems were on number theory, and have also been influential. TAKAGI (1875–1960). He proved the fundamental theorems of in pakistan abelian class field theory, as conjectured by Weber and Hilbert. NOETHER (1882–1935). Together with Artin, she laid the foundations of essay on hector from modern algebra in dissertation services, which axioms and conceptual arguments are emphasized, and she contributed to the classification of central simple algebras over number fields. HECKE (1887–1947).
Introduced HeckeL-series generalizing both Dirichlet’sL-series and Dedekind’s zeta functions. ARTIN (1898–1962). He found the “Artin reciprocity law”, which is the main theorem of class field theory (improvement of Takagi’s results). Introduced the Artin L-series. HASSE (1898–1979).
He gave the first proof of local class field theory, proved the Hasse (local-global) principle for all quadratic forms over number fields, and contributed to the classification of central simple algebras over number fields. BRAUER (1901–1977). Defined the Brauer group, and contributed to the classification of substances in central simple algebras over number fields. WEIL (1906–1998). Defined the Weil group, which enabled him to give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (1909–84). The main statements of class field theory are purely algebraic, but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. With his introduction of ide?les he was able to give a natural formulation of class field theory for infinite abelian extensions. IWASAWA (1917–1998). He introduced an important new approach into algebraic number theory which was suggested by the theory of curves over finite fields.
TATE (1925– ). Dissertation Services In Pakistan. He proved new results in group cohomology, which allowed him to give an elegant reformulation of class field theory. With Lubin he found an explicit way of generating abelian extensions of local fields. LANGLANDS (1936– ). The Langlands program2 is a vast series of conjectures that, among other things, contains a nonabelian class field theory. 2Not to be confused with its geometric analogue, sometimes referred to as the geometric Langlands pro- gram, which appears to lack arithmetic significance. Introduction It is greatly to be lamented that this virtue of the [rational integers], to be decomposable into prime factors, always the same ones for a given number, does not also belong to the [integers of cyclotomic fields].
Kummer 1844 (as translated by Andre? Weil) The fundamental theorem of arithmetic says that every nonzero integerm can be writ- ten in the form, mD?p1 pn; pi a prime number, and that this factorization is essentially unique. Consider more generally an integral domain A. An element a 2A is said to be a unit if. it has an inverse in A (element b such that ab D 1D ba). I write A for the multiplicative group of units in A. An element of A is said to prime if it is neither zero nor a unit, and if. If A is a principal ideal domain, then every nonzero element a of A can be written in the form, aD u1 n; u a unit; i a prime element; and this factorization is analytical, unique up to order and replacing each i with an associate, i.e., with its product with a unit. Our first task will be to discover to dissertation writing, what extent unique factorization holds, or fails to hold, in number fields. Three problems present themselves.
First, factorization in a field only makes sense with respect to a subring, and so we must define the “ring of integers” OK in our number field K. Secondly, since unique factorization will fail in general, we shall need to find a way of measuring by how much it fails. Finally, since factorization is only considered up to units, in order to fully understand the arithmetic of K, we need to understand the structure of the group of units UK in OK . THE RING OF INTEGERS. Let K be an algebraic number field. Each element ? of K satisfies an equation. ?nCa1? n?1 C Ca0 D 0. with coefficients a1; : : : ;an in Q, and ? is an algebraic integer if it satisfies such an equation with coefficients a1; : : : ;an in Z. We shall see that the algebraic integers form a subring OK of K. The criterion as stated is difficult to apply. We shall show (2.11) that ? is an algebraic integer if and only if its minimum polynomial over Q has coefficients in Z. Consider for example the field K D Q? p d?, where d is a square-free integer. The. minimum polynomial of ? D aCb p d , b ¤ 0, a;b 2Q, is. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and so ? is an algebraic integer if and only if. 2a 2 Z; a2?b2d 2 Z: From this it follows easily that, when d 2;3 mod 4, ? is an algebraic integer if and only if a and b are integers, i.e., and, when d 1 mod 4, ? is an algebraic integer if and only if a and b are either both integers or both half-integers, i.e., For example, the minimum polynomial of 1=2C p 5=2 is X2?X ?1, and so 1=2C. is an algebraic integer in Q? p 5?. Let d be a primitive d th root of 1, for essay from, example, d D exp.2i=d/, and letK DQ?d ?. Then we shall see (6.2) that. OK D Z?d ?D ?P.
as one would hope. A nonzero element of an integral domain A is said to be irreducible if it is not a unit, and can’t be written as a product of writing two nonunits. For example, a prime element is on hector the iliad, (obviously) irreducible. A ring A is a unique factorization domain if every nonzero element of A can be expressed as a product of irreducible elements in essentially one way. Is the ring of integers OK a unique factorization domain?
No, not in general! We shall see that each element of OK can be written as a product of irreducible elements (this is true for all Noetherian rings), and so it is the uniqueness that fails. For example, in Z? p ?5? we have. 6D 2 3D .1C p ?5/.1?. To see that 2, 3, 1C p ?5, 1?. p ?5 are irreducible, and no two are associates, we use the. p ?5 7! a2C5b2: This is multiplicative, and it is easy to see that, for ? 2OK , Nm.?/D 1 ” ? N? D 1 ” ? is a unit. (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. p ?5/D 6. Services In Pakistan. Thus Nm.?/D 1;2;3, or 6. In the. first case, ? is a unit, the second and third cases don’t occur, and in the fourth case ? is a unit. A similar argument shows that 2;3, and 1?. p ?5 are irreducible. Designer Cover Letter Ideas. Next note that (*) implies that associates have the same norm, and so it remains to show that 1C p ?5 and.
1? p ?5 are not associates, but. has no solution with a;b 2 Z. Why does unique factorization fail in OK? The problem is that irreducible elements in. Services In Pakistan. OK need not be prime. In the above example, 1C p ?5 divides 2 3 but it divides neither 2. Analytical Rubrics. nor 3. In fact, in an integral domain in which factorizations exist (e.g. a Noetherian ring), factorization is unique if all irreducible elements are prime. What can we recover? Consider. 210D 6 35D 10 21: If we were naive, we might say this shows factorization is not unique in Z; instead, we recognize that there is a unique factorization underlying these two decompositions, namely, The idea of Kummer and Dedekind was to enlarge the set of “prime numbers” so that, for example, in dissertation writing in pakistan, Z? p ?5? there is a unique factorization, 6D .p1 p2/.p3 p4/D .p1 p3/.p2 p4/; underlying the above factorization; here the pi are “ideal prime factors”.
How do we define “ideal factors”? Clearly, an ideal factor should be characterized. by the algebraic integers it divides. Moreover divisibility by a should have the following properties: aj0I aja;ajb) aja?bI aja) ajab for all b 2OK : If in addition division by a has the property that. ajab) aja or ajb; then we call a a “prime ideal factor”. Since all we know about an ideal factor is the set of elements it divides, we may as well identify it with this set. Thus an ideal factor a is a set of elements of OK such that. 0 2 aI a;b 2 a) a?b 2 aI a 2 a) ab 2 a for all b 2OK I. it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of you will recognize that an ideal factor is what we now call an ideal, and a prime ideal factor is a prime ideal. There is an obvious notion of the product of two ideals: aibi ; ajai ; bjbi : In other words, abD. nX aibi j ai 2 a; bi 2 b. One see easily that this is again an ideal, and that if. aD .a1; . ;am/ and bD .b1; . ;bn/ then a bD .a1b1; . ;aibj ; . ;ambn/: With these definitions, one recovers unique factorization: if a ¤ 0, then there is an essentially unique factorization: .a/D p1 pn with each pi a prime ideal. In the above example, .6/D .2;1C p ?5/.2;1?.
In fact, I claim. .2;1C p ?5/.2;1?. .3;1C p ?5/.3;1?. .2;1? p ?5/.3;1?. For example, .2;1C p ?5/.2;1?. p ?5;6/. Since every gen- erator is divisible by 2, we see that. Essay. .2;1C p ?5/.2;1?. Conversely, 2D 6?4 2 .4;2C2. and so .2;1C p ?5/.2;1?. p ?5/ D .2/, as claimed. I further claim that the four ideals. .2;1C p ?5/, .2;1?. Dissertation Writing Services In Pakistan. p ?5/, and .3;1?. p ?5/ are all prime.
For example, the obvious map Z! Z? p ?5?=.3;1?. p ?5/ is surjective with kernel .3/, and so. Z? p ?5?=.3;1?. which is an integral domain. How far is this from what we want, namely, unique factorization of elements? In other. words, how many “ideal” elements have we had to add to our “real” elements to get unique factorization. In a certain sense, only a finite number: we shall see that there exists a finite set S of ideals such that every ideal is of the form a .a/ for some a 2 S and some a 2OK . Better, we shall construct a group I of “fractional” ideals in which the principal fractional ideals .a/, a 2K, form a subgroup P of finite index.
The index is called the class number hK of K. We shall see that. hK D 1 ” OK is a principal ideal domain ” OK is a unique factorization domain. Unlike Z, OK can have infinitely many units. For example, .1C p 2/ is a unit of infinite. order in Z? p 2? W. p 2/m ¤ 1 if m¤ 0: In fact Z? p 2? D f?.1C. p 2/m jm 2 Zg, and so. Z? p 2? f?1gffree abelian group of essay the iliad rank 1g: In general, we shall show (unit theorem) that the roots of 1 in K form a finite group .K/, and that. OK .K/Z r (as an abelian group); moreover, we shall find r: One motivation for the development of algebraic number theory was the attempt to prove Fermat’s last “theorem”, i.e., when m 3, there are no integer solutions .x;y;z/ to the equation. with all of x;y;z nonzero. WhenmD 3, this can proved by the method of “infinite descent”, i.e., from one solution, you show that you can construct a smaller solution, which leads to a contradiction3. The proof makes use of the factorization. Y 3 DZ3?X3 D .Z?X/.Z2CXZCX2/; and it was recognized that a stumbling block to proving the theorem for larger m is that no such factorization exists into polynomials with integer coefficients of degree 2. This led people to look at more general factorizations. In a famous incident, the French mathematician Lame? gave a talk at the Paris Academy in 1847 in which he claimed to prove Fermat’s last theorem using the following ideas. Let p 2 be a prime, and suppose x, y, z are nonzero integers such that.
Write xp D zp?yp D. Y .z? iy/; 0 i p?1; D e2i=p: He then showed how to obtain a smaller solution to the equation, and hence a contradiction. Liouville immediately questioned a step in Lame?’s proof in which he assumed that, in order to show that each factor .z ? iy/ is writing, a pth power, it suffices to show that the factors are relatively prime in pairs and their product is a pth power. In fact, Lame? couldn’t justify his step (Z?? is not always a principal ideal domain), and Fermat’s last theorem was not proved for almost 150 years. Essay On Hector From. However, shortly after Lame?’s embarrassing lecture, Kummer used his results on the arithmetic of the fields Q?? to prove Fermat’s last theorem for all regular primes, i.e., for all primes p such that p does not divide the class number of Q?p?. Another application is to finding Galois groups. Dissertation. The splitting field of a polynomial f .X/ 2Q?X? is a Galois extension of Q. In a basic Galois theory course, we learn how to compute the Galois group only when the from the iliad degree is very small. By using algebraic number theory one can write down an algorithm to do it for writing services in pakistan, any degree. For applications of algebraic number theory to elliptic curves, see, for example, Milne 2006. Some comments on the literature. COMPUTATIONAL NUMBER THEORY.
Cohen 1993 and Pohst and Zassenhaus 1989 provide algorithms for most of the construc- tions we make in this course. The first assumes the reader knows number theory, whereas the second develops the whole subject algorithmically. Cohen’s book is the more useful as a supplement to designer cover letter ideas, this course, but wasn’t available when these notes were first written. While the dissertation writing in pakistan books are concerned with more-or-less practical algorithms for fields of small degree and small discriminant, Lenstra (1992) concentrates on finding “good” general algorithms. 3The simplest proof by infinite descent is that showing that p 2 is irrational. HISTORY OF ALGEBRAIC NUMBER THEORY.
Dedekind 1996, with its introduction by Stillwell, gives an excellent idea of i like the poetry of sylvia essay how algebraic number theory developed. Edwards 1977 is a history of algebraic number theory, con- centrating on the efforts to prove Fermat’s last theorem. The notes in Narkiewicz 1990 document the origins of most significant results in algebraic number theory. Lemmermeyer 2009, which explains the origins of “ideal numbers”, and dissertation writing services, other writings by the same author, e.g., Lemmermeyer 2000, 2007. 0-1 Let d be a square-free integer. Complete the verification that the ring of cover ideas integers in Q? p d? is as described. 0-2 Complete the verification that, in Z? p ?5?, .6/D .2;1C p ?5/.2;1?. is a factorization of .6/ into a product of prime ideals. CHAPTER 1 Preliminaries from Commutative.
Many results that were first proved for dissertation services, rings of integers in number fields are true for more general commutative rings, and it is more natural to prove them in that context.1. All rings will be commutative, and have an essay from, identity element (i.e., an element 1 such that 1a D a for all a 2 A), and a homomorphism of services rings will map the compare essays identity element to the identity element. A ring B together with a homomorphism of rings A! B will be referred to as an A-algebra. We use this terminology mainly when A is a subring of B . In this case, for elements ?1; . ;?m of B , A??1; . Writing Services. ;?m? denotes the smallest subring of B containing A and the ?i . It consists of substances all polynomials in the ?i with coefficients in A, i.e., elements of the form X. ai1. im? i1 1 . ? im m ; ai1. im 2 A: We also refer to A??1; . ;?m? as the A-subalgebra of B generated by the ?i , and when B D A??1; . ;?m? we say that the ?i generate B as an A-algebra. For elements a1;a2; : : : of A, we let .a1;a2; : : :/ denote the smallest ideal containing the ai . It consists of finite sums. P ciai , ci 2 A, and it is called the ideal generated by. a1;a2; : : :. When a and b are ideals in A, we define. aCbD faCb j a 2 a, b 2 bg: It is again an ideal in A — in fact, it is the smallest ideal containing both a and b. If aD .a1; . ;am/ and bD .b1; . ;bn/, then aCbD .a1; . ;am;b1; . ;bn/: Given an ideal a in A, we can form the quotient ring A=a. Let f WA!
A=a be the homomorphism a 7! aCa; then b 7! f ?1.b/ defines a one-to-one correspondence between the ideals of A=a and the ideals of A containing a, and. 1See also the writing notes A Primer of Commutative Algebra available on my website. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. A proper ideal a of A is prime if ab 2 a) a or b 2 a. An ideal a is prime if and only if the quotient ring A=a is an integral domain. A nonzero element of A is said to be prime if ./ is a prime ideal; equivalently, if jab) ja or jb. An ideal m in A is maximal if it is maximal among the proper ideals of A, i.e., if m¤A and there does not exist an ideal a ¤ A containing m but distinct from essays it. An ideal a is maximal if and only if A=a is a field. Every proper ideal a of A is contained in dissertation writing, a maximal ideal — if A is Noetherian (see below) this is obvious; otherwise the proof requires Zorn’s lemma.
In particular, every nonunit in A is contained in a maximal ideal. There are the implications: A is a Euclidean domain) A is a principal ideal domain ) A is a unique factorization domain (see any good graduate algebra course). Ideals in products of rings. PROPOSITION 1.1 Consider a product of rings AB . If a and rubrics, b are ideals in A and B respectively, then ab is an dissertation writing services in pakistan, ideal in AB , and every ideal in AB is of this form. The prime ideals of AB are the ideals of the analytical rubrics form.
pB (p a prime ideal of A), Ap (p a prime ideal of B). PROOF. Let c be an ideal in AB , and let. aD fa 2 A j .a;0/ 2 cg; bD fb 2 B j .0;b/ 2 cg: Clearly a b c. Conversely, let .a;b/ 2 c. Then .a;0/ D .a;b/ .1;0/ 2 c and .0;b/ D .a;b/ .0;1/ 2 c, and so .a;b/ 2 ab: Recall that an ideal c C is prime if and only if C=c is an writing services in pakistan, integral domain. The map. Compare And Contrat. has kernel ab, and hence induces an isomorphism. Now use that a product of rings is an integral domain if and only if one ring is zero and the other is an integral domain. 2. REMARK 1.2 The lemma extends in an obvious way to a finite product of rings: the ideals in A1 Am are of the form a1 am with ai an ideal in Ai ; moreover, a1 am is prime if and only if there is a j such that aj is a prime ideal in Aj and ai DAi for i ¤ j: A ring A is Noetherian if every ideal in A is finitely generated. PROPOSITION 1.3 The following conditions on a ring A are equivalent: (a) A is Noetherian. (b) Every ascending chain of ideals. eventually becomes constant, i.e., for some n, an D anC1 D . (c) Every nonempty set S of services in pakistan ideals in A has a maximal element, i.e., there exists an ideal in S not properly contained in any other ideal in S . PROOF. (a) (b): Let a D S. ai ; it is an phoenix, ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and so an D anC1 D D a. (b) (c): Let a1 2 S . If a1 is not a maximal element of S , then there exists an a2 2 S such that a1 a2. If a2 is not maximal, then there exists an a3 etc.. From (b) we know that this process will lead to a maximal element after only finitely many steps. (c) (a): Let a be an ideal in A, and let S be the set of finitely generated ideals contained in a. Then S is nonempty because it contains the zero ideal, and so it contains a maximal element, say, a0 D .a1; : : : ;ar/.
If a0 ¤ a, then there exists an element a 2 ar a0, and .a1; : : : ;ar ;a/ will be a finitely generated ideal in a properly containing a0. This contradicts the definition of a0. 2. A famous theorem of Hilbert states that k?X1; . ;Xn? is Noetherian. In practice, al- most all the rings that arise naturally in algebraic number theory or algebraic geometry are Noetherian, but not all rings are Noetherian. For example, the ring k?X1; : : : ;Xn; : : :? of polynomials in an infinite sequence of symbols is not Noetherian because the writing services chain of ideals. Substances. never becomes constant. PROPOSITION 1.4 Every nonzero nonunit element of a Noetherian integral domain can be written as a product of irreducible elements.
PROOF. We shall need to use that, for elements a and b of an integral domain A, .a/ .b/ ” bja, with equality if and only if b D aunit: The first assertion is obvious. For the second, note that if a D bc and b D ad then a D bc D adc, and so dc D 1. Hence both c and d are units. Suppose the statement of the proposition is false for dissertation writing, a Noetherian integral domain A. Then there exists an element a 2 A which contradicts the statement and is such that .a/ is maximal among the ideals generated by such elements (here we use that A is Noetherian). Since a can not be written as a product of irreducible elements, it is not itself irreducible, and essay, so a D bc with b and c nonunits.
Clearly .b/ .a/, and the ideals can’t be equal for otherwise c would be a unit. From the maximality of .a/, we deduce that b can be written as a product of irreducible elements, and similarly for c. Thus a is a product of irreducible elements, and we have a contradiction. 2. REMARK 1.5 Note that the proposition fails for the ring O of all algebraic integers in the algebraic closure of Q in C, because, for example, we can keep in extracting square roots — an algebraic integer ? can not be an irreducible element of dissertation writing O because. p ? will also be. an algebraic integer and ? D p ? p ?. Thus O is not Noetherian. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Let A be a ring. An A-module M is said to be Noetherian if every submodule is finitely generated.
PROPOSITION 1.6 The following conditions on an A-module M are equivalent: (a) M is Noetherian; (b) every ascending chain of submodules eventually becomes constant; (c) every nonempty set of submodules in M has a maximal element. PROOF. And Contrat Essays. Similar to dissertation services in pakistan, the proof of Proposition 1.3. 2. PROPOSITION 1.7 Let M be an A-module, and let N be a submodule of M . If N and M=N are both Noetherian, then so also is M . PROOF. I claim that if M 0 M 00 are submodules of M such that M 0N DM 00N and M 0 and M 00 have the same image in M=N , then M 0 DM 00. To see this, let x 2M 00; the second condition implies that there exists a y 2M 0 with the same image as x inM=N , i.e., such that x?y 2N . Then x?y 2M 00N M 0, and so x 2M 0. Now consider an ascending chain of submodules of M . If M=N is Noetherian, the image of the chain in M=N becomes constant, and if N is Noetherian, the intersection of the chain with N becomes constant. Now the claim shows that the chain itself becomes constant. 2. PROPOSITION 1.8 Let A be a Noetherian ring.
Then every finitely generated A-module is Noetherian. PROOF. If M is generated by a single element, then M A=a for prescribing in, some ideal a in A, and the statement is obvious. We argue by induction on the minimum number n of generators ofM . Services In Pakistan. SinceM contains a submoduleN generated by n?1 elements such that the quotient M=N is generated by a single element, the statement follows from (1.7). 2. A ring A is said to local if it has exactly one maximal ideal m. In this case, A D Arm (complement of m in A). LEMMA 1.9 (NAKAYAMA’S LEMMA) Let A be a local Noetherian ring, and let a be a proper ideal in A. Let M be a finitely generated A-module, and define. aM D f P aimi j ai 2 a; mi 2M g : (a) If aM DM , then M D 0: (b) If N is a submodule of M such that N CaM DM , then N DM: Rings of fractions. PROOF. Substances In. (a) Suppose that aM D M but M ¤ 0. Choose a minimal set of generators fe1; : : : ; eng for M , n 1, and write. e1 D a1e1C Canen, ai 2 a: Then .1?a1/e1 D a2e2C Canen: As 1? a1 is not in m, it is a unit, and so fe2; . Dissertation Writing In Pakistan. ; eng generates M , which contradicts our choice of and contrat essays fe1; : : : ; eng. (b) It suffices to show that a.M=N/DM=N for then (a) shows that M=N D 0. Con- sider mCN , m 2M . From the assumption, we can write. aimi , with ai 2 a, mi 2M: and so mCN 2 a.M=N/: 2. The hypothesis that M be finitely generated in the lemma is essential. For example, if A is a local integral domain with maximal ideal m ¤ 0, then mM DM for any field M containing A but M ¤ 0. Rings of fractions.
Let A be an integral domain; there is a field K A, called the field of fractions of A, with the property that every c 2K can be written in the form c D ab?1 with a;b 2A and b ¤ 0. For example, Q is the field of fractions of dissertation writing services Z, and k.X/ is the field of fractions of k?X?: Let A be an integral domain with field of fractions K. A subset S of A is said to be multiplicative if 0 … S , 1 2 S , and S is closed under multiplication. If S is a multiplicative subset, then we define. S?1AD fa=b 2K j b 2 Sg: It is obviously a subring of K: EXAMPLE 1.10 (a) Let t be a nonzero element of A; then. St def D f1,t ,t2. g. is a multiplicative subset of A, and we (sometimes) write At for S?1t A. For example, if d is a nonzero integer, then2 Zd consists of those elements of Q whose denominator divides some power of d : Zd D fa=dn 2Q j a 2 Z, n 0g: (b) If p is a prime ideal, then SpDArp is a multiplicative set (if neither a nor b belongs to p, then ab does not belong to p/. We write Ap for S?1p A. For example, Z.p/ D fm=n 2Q j n is not divisible by pg:
2This notation conflicts with a later notation in which Zp denotes the ring of letter p-adic integers. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. PROPOSITION 1.11 Consider an services in pakistan, integral domainA and a multiplicative subset S ofA. For an ideal a of A, write ae for the ideal it generates in S?1A; for an ideal a of S?1A, write ac for aA. Then: ace D a for essay the iliad, all ideals a of S?1A aec D a if a is a prime ideal of A disjoint from S: PROOF. Let a be an ideal in S?1A. Dissertation Writing Services In Pakistan. Clearly .aA/e a because aA a and a is an ideal in S?1A. For the reverse inclusion, let b 2 a. We can write it b D a=s with a 2 A, s 2 S . Then aD s .a=s/ 2 aA, and so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . Clearly .S?1p/A p. For the reverse inclu- sion, let a=s 2 .S?1p/A, a 2 p, s 2 S . Consider the equation a. s s D a 2 p. Both a=s. and s are in A, and so at least one of a=s or s is in p (because it is prime); but s … p (by assumption), and so a=s 2 p: 2. PROPOSITION 1.12 Let A be an integral domain, and let S be a multiplicative subset of A. The map p 7! pe defD p S?1A is a bijection from the set of prime ideals in A such that pS D? to the set of prime ideals in S?1A; the inverse map is p 7! pA. PROOF. It is easy to see that. p a prime ideal disjoint from S) pe is a prime ideal in S?1A, p a prime ideal in S?1A) pA is a prime ideal in A disjoint from S; and (1.11) shows that the two maps are inverse.
2. EXAMPLE 1.13 (a) If p is of sylvia essay, a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). (b) We list the prime ideals in some rings: Note that in writing, general, for t a nonzero element of an integral domain, fprime ideals of Atg $ fprime ideals of compare and contrat A not containing tg. fprime ideals of A=.t/g $ fprime ideals of A containing tg: The Chinese remainder theorem. Recall the classical form of the theorem: let d1; . ;dn be integers, relatively prime in pairs; then for any integers x1; . ;xn, the congruences. The Chinese remainder theorem. have a simultaneous solution x 2 Z; moreover, if x is one solution, then the other solutions are the integers of the form xCmd with m 2 Z and d D. We want to translate this in terms of ideals. Integersm and n are relatively prime if and only if .m;n/D Z, i.e., if and only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to be relatively prime if aCbD A. If m1; . Writing Services In Pakistan. ;mk are integers, then T .mi / D .m/ where m is the least common multiple. of the mi . Thus T .mi / . Q mi /, which equals. Q .mi /. If the mi are relatively prime in. pairs, then mD Q mi , and so we have. Q .mi /. Note that in general, a1 a2 an a1a2 . an; but the two ideals need not be equal. These remarks suggest the following statement. I Like Essay. THEOREM 1.14 Let a1; . ;an be ideals in a ring A, relatively prime in pairs. Then for any elements x1; . ;xn of A, the congruences. have a simultaneous solution x 2 A; moreover, if x is one solution, then the other solutions are the writing in pakistan elements of the form xC a with a 2. Q ai . In other words, the. natural maps give an exact sequence.
PROOF. Suppose first that n D 2. As a1C a2 D A, there are elements ai 2 ai such that a1Ca2 D 1. The element x D a1x2Ca2x1 has the required property. Prescribing. For each i we can find elements ai 2 a1 and bi 2 ai such that. ai Cbi D 1, all i 2: The product Q i2.ai Cbi /D 1, and lies in a1C. Q i2 ai , and so. We can now apply the theorem in the case nD 2 to obtain an element y1 of A such that. y1 1 mod a1; y1 0 mod Y. These conditions imply. y1 1 mod a1; y1 0 mod aj , all j 1: Similarly, there exist elements y2; . ;yn such that. yi 1 mod ai ; yi 0 mod aj for j ¤ i: The element x D P xiyi now satisfies the requirements. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. It remains to prove that T. ai . We have already noted that T. ai . First suppose that nD 2, and let a1Ca2 D 1, as before. For c 2 a1a2, we have. Dissertation Services. c D a1cCa2c 2 a1 a2. which proves that a1 a2 D a1a2.
We complete the proof by induction. This allows us to assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . The theorem extends to A-modules. THEOREM 1.15 Let a1; . ;an be ideals in A, relatively prime in pairs, and let M be an A-module. There is an exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to use tensor products, which I now review. Review of tensor products. Let M , N , and P be A-modules. A mapping f WM N ! P is said to be A-bilinear if. f .mCm0;n/D f .m;n/Cf .m0;n/ f .m;nCn0/D f .m;n/Cf .m;n0/ f .am;n/D af .m;n/D f .m;an/ 9=; all a 2 A; m;m0 2M; n;n0 2N: i.e., if it is linear in essay from the iliad, each variable. A pair .Q;f / consisting of an A-module Q and an A-bilinear map f WM N !Q is called the tensor product of M and N if any other A- bilinear map f 0WM N ! P factors uniquely into f 0 D ? ?f with ?WQ!
P A-linear. The tensor product exists, and is unique (up to a unique isomorphism making the obvious diagram commute). We denote it by dissertation writing services in pakistan, M ?AN , and we write .m;n/ 7! m?n for f . The pair .M ?AN;.m;n/ 7!m?n/ is characterized by each of the following two conditions: (a) The mapM N !M ?AN is A-bilinear, and any other A-bilinear mapM N ! P is of the form .m;n/ 7! ?.m?n/ for a unique A-linear map ?WM ?AN ! P ; thus. BilinA.M N;P /D HomA.M ?AN;P /: (b) TheA-moduleM?AN has as generators them?n,m2M , n2N , and as relations. 9=; all a 2 A; m;m0 2M; n;n0 2N: Tensor products commute with direct sums: there is a canonical isomorphism. Review of tensor products.
It follows that if M and N are free A-modules3 with bases .ei / and .fj / respectively, then M ?AN is a free A-module with basis .ei ? fj /. Essay On Hector The Iliad. In particular, if V and W are vector spaces over a field k of dimensions m and n respectively, then V ?kW is writing services, a vector space over k of dimension mn. Let ?WM !M 0 and ?WN !N 0 be A-linear maps. Then. .m;n/ 7! ?.m/??.n/WM N !M 0?AN 0. is A-bilinear, and therefore factors uniquely through M N !M ?AN . Thus there is a unique A-linear map ???WM ?AN !M 0?AN 0 such that. REMARK 1.16 The tensor product of two matrices regarded as linear maps is called their Kronecker product.4 If A is mn (so a linear map kn! km) and B is r s (so a linear map ks! kr ), then A?B is the mr ns matrix (linear map kns! kmr ) with. 0B@ a11B a1nB. : : : . am1B amnB. 1CA : LEMMA 1.17 If ?WM !M 0 and ?WN !N 0 are surjective, then so also is. ???WM ?AN !M 0 ?AN. PROOF. Recall that M 0?N 0 is dissertation phoenix, generated as an A-module by the elements m0?n0, m0 2 M 0, n0 2 N 0. Dissertation Writing. By assumption m0 D ?.m/ for some m 2M and n0 D ?.n/ for some n 2 N , and som0?n0 D ?.m/??.n/D .???/.m?n/. Therefore the analytical essay rubrics image of ??? contains a set of dissertation generators for M 0?AN 0 and so it is equal to it.
2. One can also show that if M 0!M !M 00! 0. is exact, then so also is. M 0?AP !M ?AP !M 00 ?AP ! 0: For example, if we tensor the exact sequence. with M , we obtain an exact sequence. a?AM !M ! .A=a/?AM ! 0 (2) 3Let M be an A-module. Elements e1; : : : ; em form a basis for M if every element of M can be expressed uniquely as a linear combination of the ei ’s with coefficients in cover ideas, A. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of A-modules, and M is said to be a free A-module of dissertation services rank m. 4Kronecker products of matrices pre-date tensor products by about 70 years.
1. Of Controlled In. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. The image of a?AM in M is. P aimi j ai 2 a, mi 2M g ; and so we obtain from the exact sequence (2) that. Writing. By way of contrast, ifM !N is injective, thenM ?AP !N ?AP need not be injective. For example, take A D Z, and note that .Z. Of Sylvia Plath. m ! Z/?Z .Z=mZ/ equals Z=mZ. which is the zero map. PROOF (OF THEOREM 1.15) Return to the situation of the theorem. When we tensor the isomorphism. with M , we get an isomorphism. M=aM ' .A=a/?AM ' ! Q .A=ai /?AM ' EXTENSION OF SCALARS.
If A! B is an A-algebra and M is an dissertation in pakistan, A-module, then B?AM has a natural structure of i like the poetry of sylvia a B-module for which. b.b0?m/D bb0?m; b;b0 2 B; m 2M: We say that B?AM is the B-module obtained from M by extension of scalars. The map m 7! 1?mWM ! B ?AM has the following universal property: it is A-linear, and for any A-linear map ?WM ! N from M into a B-module N , there is writing in pakistan, a unique B-linear map ?0WB?AM !N such that ?0.1?m/D ?.m/. Thus ? 7! ?0 defines an isomorphism. Essay. HomA.M;N /! HomB.B?AM;N/, N a B-module: For example, A?AM DM . If M is dissertation writing in pakistan, a free A-module with basis e1; : : : ; em, then B?AM is a free B-module with basis 1? e1; : : : ;1? em. TENSOR PRODUCTS OF ALGEBRAS.
If f WA! B and gWA! C are A-algebras, then B ?A C has a natural structure of an A-algebra: the product structure is determined by the rule. .b? c/.b0? c0/D bb0? cc0. and the map A! B?AC is a 7! f .a/?1D 1?g.a/. For example, there is a canonical isomorphism. a?f 7! af WK?k k?X1; : : : ;Xm?!K?X1; : : : ;Xm? (4) Review of tensor products. TENSOR PRODUCTS OF FIELDS. We are now able to prescribing of controlled substances, compute K?k? if K is a finite separable field extension of a field k and ? is an arbitrary field extension of k. According to the primitive element theorem (FT 5.1), K D k??? for some ? 2K. Let f .X/ be the minimum polynomial of dissertation writing ?. By definition this means that the map g.X/ 7! g.?/ determines an isomorphism.
Hence K?k? ' .k?X?=.f .X///?k? '??X?=.f .X// by (3) and (4). Because K is separable over k, f .X/ has distinct roots. Therefore f .X/ factors in ??X? into monic irreducible polynomials. that are relatively prime in pairs. We can apply the Chinese Remainder Theorem to deduce that. Finally, ??X?=.fi .X// is prescribing, a finite separable field extension of ? of degree degfi . Thus we have proved the following result: THEOREM 1.18 Let K be a finite separable field extension of k, and let ? be an arbitrary field extension. Then K?k? is a product of finite separable field extensions of ?, If ? is a primitive element for K=k, then the image ?i of ? in ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the minimum polynomials for ? and in pakistan, ?i respectively, then. EXAMPLE 1.19 Let K DQ??? with ? algebraic over Q. Then. C?QK ' C?Q .Q?X?=.f .X///' C?X?=..f .X//' Yr. iD1 C?X?=.X ??i / Cr : Here ?1; : : : ;?r are the conjugates of ? in C. The composite of ? 7! 1??WK!
C?QK with projection onto the i th factor is. We note that it is essential to assume in (1.18) that K is separable over k. If not, there will be an designer letter ideas, ? 2K such that ?p 2 k but ? … k, and the ring K?kK will contain an element ? D .??1?1??/¤ 0 such that. ?p D ?p?1?1??p D ?p.1?1/??p.1?1/D 0: Hence K?kK contains a nonzero nilpotent element, and so it can’t be a product of fields. NOTES Ideals were introduced and studied by Dedekind for rings of algebraic integers, and later by others in polynomial rings. Services In Pakistan. It was not until the 1920s that the theory was placed in its most natural setting, that of arbitrary commutative rings (by Emil Artin and Emmy Noether). Embed this document on compare and contrat, your website. If you don't receive any email, please check your Junk Mail box.
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Fast Food Restaurant Business Plan. Nudlez, a Washington Corporation, is in the fast-food service industry. The company has developed a unique business model that reduces overhead by utilizing mobile vending units as opposed to brick and mortar stores. Dissertation Writing! This is editor university a very popular alternative to dining in larger business centric cities. Nudlez provides high quality, fast-served Asian-styled meals, based on a central theme of dissertation in pakistan, noodles. The meals cater for variety and are delivered fresh faster than other noodle-based products. The meals are provided to the consumer initially through mobile Noodle Vending Units (NVU's), which provide widespread point of sale coverage.
The NVU's are state-of-the-art, complying with health regulations. The first trial NVU was tested in Seattle in July 20XX, receiving great response. The NVU will become a product in itself, as Nudlez expands rapidly through franchising. Nudlez is a globally transportable business. Valued in 20XX at $105 billion, the domestic market for fast food is large and segmented, with Asian style foods representing 1 of the top 3 choices in Seattle at essay 35% of the market. Dissertation Writing! Nudlez will initially target central business district (CBD) lunchtime crowds, where high-income earning but time-poor individuals are most greatly concentrated. Shopping malls, university campuses, weekend markets, and local business areas will then be targeted. Event catering and supermarket convenience foods will be markets that Nudlez then diversifies into. Nudlez will build brand recognition through its Noodle Vending Units being placed strategically throughout large cities' business districts.
Once brand recognition has been achieved, catering services will be offered, and the Nudlez products will be available in supermarkets. Additionally, Nudlez offers franchise opportunities to help further expand its reach, along with adding an additional revenue stream. The owners have significant experience in: the hospitality industry, advertising, and business management. Letter! The founder previously owned two very successful Thai restaurants in Seattle but grew tired of dissertation writing in pakistan, paying exorbitant rent hoping customers would turn up each night. His Nudlez concept takes the product to the street! Nudlez has strong financials, and is always cash flow positive. Designer Letter Ideas! Revenues are projected to grow to nearly $58,000,000 by Year 5, with EBITDA totaling $15,000,000. An initial analysis indicates that only dissertation writing in pakistan, 100 meals per day per NVU must be sold to break even. The strong financial plan ensures that the company will be attractive as an acquisition for in, exit. The purpose of this business plan is to raise $1,500,000 from an investor, in the form of a convertible and redeemable note, providing a 32% compound annual return. Nudlez is committed to dissertation services in pakistan the rollout of its 'wok-fresh' fast food business in 20XX.
Nudlez is trademark registered. Nudlez Inc. was incorporated in the State of Washington in March 20XX. Nudlez is currently equally owned by Mr. Dan Billings and Mr. Editor University! Bill Cook. They have already spent over $75,000 in direct costs of product testing and development, and dissertation services will commit a further $200,000 in 20XX, prior to funds being required from the investor. Both owners will each hold two voting rights on the Board with the investor holding one. After running two successful Thai restaurants in Seattle for five years, Dan Billings spent most of the past two years in university phoenix Thailand looking at business opportunities. While living there he ate almost exclusively as the locals did, from services in pakistan street vendor food outlets. Not only and contrat essays, was the food freshly cooked in front of him, but the dissertation services, visual entertainment combined to augment the vibrant street culture. His initial thoughts were, this is what's missing from our staid cities in the United States.
Using this as his motivator he began to realize that there were also major commercial benefits to running a fast-food network where the majority of outlets were not restricted to one location. In! For a start, shop fronts in good locations are expensive. In Pakistan! The key to making such a venture successful domestically would be to ensure that the plath essay, vendor outlets are very hygienic, comply with health standards, serviced by an efficient re-supply network producing consistent meals, with professionalism and quality paramount. He spent considerable time designing a purpose built Noodle Vending Unit. He also moved back to Seattle and renewed his ties in the culinary community to elicit the design of fresh and tasty meals to be served from the NVU. The Nudlez brand was born! Realizing the dissertation writing, opportunity to seize a market niche through rapid expansion, the inventor also joined forces with an experienced international business team to develop and implement that initiative. During the essay on hector from the iliad, start-up phase of the business, food sales revenue will come from NVU lunchtime customers in busy Seattle CBD locations. A small office will be setup in Seattle that will serve as the headquarters of the services in pakistan, company. The principal product of Nudlez is fast-served, quality, Asian-styled individual meals.
The meals are based on a central theme of noodles, but have a flexible degree of from the iliad, customization to suit the purchasers' tastes. The second, but equally important supporting product, is the mobile cooking unit used to prepare the individual meals. This Noodle Vending Unit (NVU) has been specifically designed to dissertation services in pakistan produce the Nudlez meals in the shortest time possible. While Nudlez will own and analytical essay operate some NVU's, the bulk of the fleet will be purchased and operated by franchisees. These two products symbiotically solidify the Nudlez brand name. Nudlez is a state-of-the-art stir-fry noodle street vending system. Dissertation! While Nudlez's footprint in the three major metropolitan cities in Washington will be underpinned by the NVU fleet and a smaller number of strategically placed Nudlez stores, the compare and contrat, aim in year-3 is to diversify the revenue stream by selling the meal product alone in supermarkets for home preparation and consumption. Fast Affordable Meals.
The NVU operator cooks the meals in front of the customer within sixty seconds, using fresh ingredients. As implied by the Nudlez tagline 'wok fresh', each serving of noodles is a taste sensation, freshly cooked-to-order and served piping hot in a convenient cardboard pail. Customers are invited to create their own noodle dish following three easy steps: Step 1: Select Noodle Type; Step 2: Add Sauce; Step 3: Choose Protein. Compared to the meals offered by in pakistan, competitors, Nudlez offers a fresh, tasty, nutritious alternative. Nudlez may be slower than some fast food from the point of order, but it is significantly fresher, more nutritious, and almost 50% more profitable. One secret to the speed of designer cover, Nudlez meals is that the meats and vegetables are pre-blanched and hygienically sealed in the same pail that the dissertation, customer eventually receives.
The operator simply removes from the fridge, and combines with noodles and prescribing in sauce in the wok. Noodle Vending Unit. Each NVU is completely self-contained, mounted on wheels and totally mobile. It is delivered on-site each day by a custom designed mini-van and simply wheeled into position. Stabilizing legs are deployed, the roof pops up and it is ready to writing in pakistan go!
At the end of the day it is simply wheeled back into the mini-van and returned to base for cleaning and waste removal. Dissertation Editor University Phoenix! Ensuring the speed of the NVU operation is the Nudle-Rac. The Nudle-Rac is essentially a lightweight grid that is packed in services in pakistan the meal production facility with the various meal options. In! The Nudle-Rac is self-leveling and designed to dissertation writing services in pakistan fit perfectly in the NVU's fridge. Each NVU cart is immaculately presented in stainless steel and branded in strong colorful graphics. At night the noodle stand benefits from its own internal lighting. Construction is of the plath, highest quality kitchen-grade stainless steel.
It is designed to be versatile for dissertation writing, use in just about i like the poetry of sylvia plath essay, any place where it is legal to sell food from a temporary stall. In comparison to standard 'caravan-type' vendor units, the NVU has major advantages in that it is: cheaper to produce, smaller size for writing in pakistan, better site access, self-sustaining power (no mains required short-term), easily moved by one person, unique look, hygienic and cheap to service. The range of competitors here spans the market dominants like McDonalds, to sandwich and sushi bars, to 'vendor' foods like kebabs. However, when looking for closer similarity to analytical essay Nudlez's offering, we arrive at writing the specialized noodle-based products provided by the poetry of sylvia plath, such companies as 'Noodlist'. A storefront operation located close to Pioneer Square in Seattle, the meal variety at 'Noodlist' is services in pakistan slightly broader. However speed of preparation is significantly slower than Nudlez, necessitating the addition of standard pre-cooked, oily and somewhat bland Asian fare to satisfy the 'time-poor' consumers. It is only this time constraint that is satisfied though, as taste and nutrition are compromised.
The lack of speed in i like of sylvia essay the fresh cooked Asian fast food market is emphasized by another competitor, 'Hot Wok', offering noodle-based meals in the Sea-Tac International Airport, where a sign on the counter informs consumers that meals may take up to thirty minutes to prepare. A captive consumer group is not being exploited here. Nudlez will change that. 'Wok in writing services a Box' is an Oregon competitor that intends to open stores in essay on hector the iliad Washington beginning in 20XX. Services In Pakistan! The company currently operates 16 stores in compare and contrat the Pacific Northwest. Services In Pakistan! Market research indicates company revenue for 20XX reached $5 million, representing a growth rate of 78 percent over the prior year. This growth is encouraging to Nudlez, which through the designer cover, use of its NVU's and writing services its other competitive advantages will gain broader exposure. The existence of analytical rubrics, such competitors makes Nudlez an attractive acquisition or merger proposition. 3.4 COMPETITIVE EDGE/BARRIERS TO ENTRY. Unique modality of writing services in pakistan, fast-food availability, with unmatched preparation times Lower capital costs and overheads and high margin unit sales through broad exposure.
Healthy, fresh, fast, quality, price competitive alternative New funky, identifiable brand, professional approach and solid management Aimed at segment of large market not currently exploited by the competition. The founders of Nudlez have invested a total of $75,000 in development of the NVU, and plan to spend an additional $200,000 over the next year on ideas development of the second generation NVU. The 'Generation-2' NVU is currently under design, maintaining versatility yet improving with the introduction of inventions such as the Nudle-Rac. With Nudlez, the term customer holds two meanings; firstly as a reference to consumers who may purchase a meal from a Nudlez point of writing in pakistan, presence, and designer letter secondly as entities with an writing services in pakistan, interest in owning a Nudlez franchise. The primary target consumer group for the first phase of Nudlez fits the profile of a medium to on hector from the iliad high-income earner, with a healthy and fresh lifestyle orientation.
The first phase of the business plan is in pakistan about growing brand awareness quickly with customers of on hector from, this profile. Shortly after growing the brand awareness, as part of the Nudlez rapid growth plans, franchisee customers looking for dissertation writing services, a self-employment opportunity providing good income relative to the hours worked will be targeted. Institutions such as university campuses will also be targeted as potential franchisees. The profile will then extend through catering to the customer who is looking for a relatively low cost, value-adding product to support private or corporate events. This customer will be served at both the supermarket (DIY) and event catering levels. With sales in excess of i like the poetry, $105 billion, the in pakistan, market for fast food is substantial, and on hector the iliad the opportunity to build and grow a new exciting product with a strong brand is real. Asian style food represented approximately 35% of this revenue. Dissertation Writing Services! Although the market is mature with heavy buyer/seller saturation, it is heavily segmented on quality, style, modality and price, presenting opportunity for niche market products such as Nudlez. Customers' perception of 'healthier' is an important value proposition.
Nudlez provides this, with the added bonus of serving the meal quickly. The idea that desirable food could be prepared in a quick fashion appeals to a sizeable portion of the market; those in tourism, people engaged in a commercial capacity, busy homemakers and students. These few examples are indicative of a large and growing collection of time poor individuals placing increasing value on every spare moment of their day. Prescribing In! Nudlez has the ability to match the ravenous need for convenience with the utility that modern, fast paced lifestyles drive. Nudlez negates boundaries of culture, age and preference, through a daily mission to satiate appetites. Today the contemporary food outlet must maintain its appeal within an increasingly health-conscious market. Worldwide the dissertation writing services in pakistan, diet continues to gain momentum and compare this means that products with less appeal to the diet-conscious will suffer reduced success. Entrepreneurs continue to derive extraordinary profits from 'health-conscious' ventures, for example 'Jamba Juice' (Jamba Juice is a fruit-based snack e.g. smoothies and drinks, which offers a fresh and healthy alternative to the major players' product. Jamba Juice expects $70 Million in revenue from its 150 stores in 20XX.) Nudlez is designed to make the most of the public's increasing nutritional conscience. The economic environment for the successful outlet will take into account the factors and writing in pakistan conditions that influence the ability of its target market to afford its product.
Consumers today, particularly in business-centric cites, have a relatively high disposable income. Nudlez will be priced competitively with existing alternatives and will strive to compare and contrat attract market share through product differentiation. Strong product supported by great tasking meals with a strong focus on quality, healthy and competitively priced meals in writing in pakistan addition to the speed of producing meals Strong brand, which has been tested before full implementation. Emphasis on a fund brand culture Market conditions are favorable with Nudlez offering new niche spark in an otherwise static market Low development risk as the product currently exists Easy, low risk growth through short time to market and short time for NVU construction as well as low cost per unit Transferable business model as a result of similarities in trends and the US culture of street style food vending. Ability to prescribing capitalize on brand and concept with expansion through franchise and other market segments Low management risk due to in pakistan experienced team and the strong diversification of skills and expertise = holistic results High gross margins on meal products, which can be maximized by i like of sylvia plath, negotiating bulk purchase agreements with wholesalers. Defendability of concept as the meal products cannot be patented; this needs to be addressed by building strong brand awareness and establishing copyright on total quality system.
Focus also on rapid growth and securing prime sites in target markets NVU restocking logistics is being addressed through NVU design, which ensures breakeven results for a day's trade without restocking. Central production facilities also being established to support restocking Site permits for street vending need to be established before implementation of writing, NVU's, but also private sites will be secured to reduce exposure to local government issues Product may be vulnerable to weaken over time, which can be minimized through product innovation and diversification. Niche-specific opportunities through monopolizing niche market by building strong brand awareness and leading the street food vending culture through being vigilant with other vending business opportunities. Low restriction to editor university phoenix NVU operating hours means there can be diversity of writing services, venues and access to a very diverse target customer group Capitalize on diversity of analytical, venues and consumer groups to promote and market expansion through event catering and franchise ownership International expansion to Europe, which has a strong culture for street food vending and writing services pre-prepared meals. Substances! Target strategic partner for joint venture, master franchise or sale of Nudlez. Local government regulations in relation to dissertation writing street vendors means permits secured have minimum life period, however change in this area is slow Vulnerability to imitation products from competitors will be reduced through rapid development of Nudlez products and brand. Short-term site permits means the Nudlez brand needs to cover letter be diversified through catering, franchise and supermarket placement Seasonal demand for NVU's meaning inconsistent returns. This can be minimized through diversifying brand through catering, franchising and a supermarket product line. To forge a new, vibrant, high quality brand in the fast-food sector; to challenge the established market by offering a fresh, tasty and professional alternative; and to build a robust business model founded on high-yield, low capital outlay products. It is our vision that Nudlez will expand predominantly through franchising domestically and then to appropriate international markets.
It will diversify through catering and product placement in supermarkets. The possibility of trade acquisition or merger also exists. Given that the United States is positioned 1st in the top 10 global markets for weekly fast food consumption, the marketing strategy is based on penetrating the services in pakistan, market rapidly. This will be achieved by placing the compare and contrat essays, NVU's in strategic locations where the likelihood of consuming a takeaway meal is extremely high. Expansion of numbers of dissertation services in pakistan, NVU's and and contrat essays stores through franchising will ensure that first mover advantage is services optimized. The network is analytical supported by an integrated total quality management system ensuring consistency, which will be leveraged in the marketing strategy. Capital outlay for franchisees will be low with owners earning more dollars per dissertation hour worked than other franchise options. Expected net profit after tax for franchisees is essay 20%. The marketing objective of Nudlez is to position itself so that consumers of fast food are confident they are purchasing and consuming a healthy and fresh product, enhanced by the overall delivery experience.
The marketing objectives are to: Rapidly establish a recognized brand through street vendor points of sale supported by a smaller number of strategically placed hardstand stores Position itself as a fresh, tasty and dissertation writing services in pakistan healthy alternative to traditional fast food Maintain high gross margins on meal products Capitalize on prescribing of controlled substances in the brand through expansion using franchising and product diversification, which will include catering and services supermarket placement. Nudlez's initial strategy will be focused within Washington, commencing in Seattle then within 18 months expansion into Spokane and analytical essay the Vancouver/Portland metropolis. Beginning with company owned NVU's and central production facilities, rapid expansion will then be achieved through franchising in all target areas. NVU production will be outsourced to a manufacturer in Seattle and has been estimated at $12,500 per NVU. Dissertation Writing Services In Pakistan! Each city will have company owned NVU's, shops, a central kitchen premises and support kitchens (hubs). Product Production and Preparation.
Raw product has been sourced from selected Seattle suppliers. It is a competitive quality market, with room for a number of suppliers to diversify risk. The secret to Nudlez' fast delivery to the customer from moment of order is the compare essays, blanching and writing preparation process of the raw product at the central kitchen premises and hub sites. This will be overseen by our master-chef teams, adhering to strict health regulations. Compare And Contrat! All ordering, stocking and quality controls will be managed on dissertation writing in pakistan an in-house computer based system enabling usage patterns to be monitored maintaining ordering efficiency and minimizing wastage.
A unique system for NVU and store restocking, the Nudle-Rac is currently under design. Essentially providing completely removable internal sectioning to the refrigerators to hold the essay rubrics, different meals available, it is spring-loaded so that when one meal is removed, the next meal raises to the top for easy access. At the beginning of each day, a franchisee will arrive at their support kitchen where the dissertation services in pakistan, Nudle-Rac, preloaded with the day's meals, will be loaded directly into essay from the NVU's fridge without having to remove the dissertation writing services, NVU from the van providing an efficient stocking system and of sylvia minimizing the time meals spend out of refrigeration. Dissertation In Pakistan! For large catering events, support vans will provide additional pre-stocked Nudle-Racs to onsite NVU's. It is Nudlez objective to be acquired by another fast food company within five years by posting large revenues and designer a substantial EBITDA. Additionally, merger options, as well as an dissertation writing in pakistan, IPO may be considered at that time.
6.1 ORGANIZATIONAL STRUCTURE. Nudlez will be lead by Bill Cook, CEO. He will report to the founders and investors, which will make up the board. The CEO will oversee the CFO, CIO, Food and Human Resources departments. With over 80 years of sector diverse, international strategic and operational combined experience, professionally qualified and with energetic personalities critical to prescribing of controlled in ensuring the Nudlez values, the owners and the management provide the skills to writing deliver: Bill Cook has run start-up business operations, with broad management skills Dan Billings has owned and managed 2 successful restaurants and an advertising firm David Noor is a financial wiz with broad entrepreneurial appreciation Angela Gates is dissertation phoenix not your typical CIO, with a creative flair and eye for efficiency Sheila Arch brings the important HR skills combined with a marketing bent. Bill Cook - CEO Director (Systems, Strategy Finance) Bill is a Seattle boy, born and bred, with an intimate knowledge of the writing in pakistan, cultural and niche diversities of Seattle.
Originally with an honors degree in Civil Engineering, Bill worked in the Seattle construction industry from the 80's boom time through the recession of the early 90's. From there he traveled overseas working in such places as the UK, Bosnia (during the war), Romania and Holland. The bulk of the poetry of sylvia essay, his experience in these eight years centered on strategic management of start-up ventures. These ranged from establishing critical engineering support facilities for the UN in services in pakistan Bosnia, to expending Shell's retail network across Eastern Europe, to property development ventures in Romania. Bill returned to substances Seattle in 20XX to utilize this broad strategic experience on home soil, initially focusing on the property sector. Bill has known Dan for over 20 years and quickly assessed the potential of his Nudlez idea, convincing Dan that rapid expansion was key to securing the market niche. Bill graduated his MBA with High Distinction in 20XX. Dan Billings - Director (Marketing Food Technology) After a successful career in in pakistan advertising culminating in analytical essay the establishment of his own Seattle based agency, Dan entered the hospitality industry.
His first venture, Pad Real Thai, a Thai restaurant and takeaway situated on bustling Pine Street, was an instant hit. Writing In Pakistan! An innovative and analytical essay funky menu, supported with clever marketing (including the use of real Bangkok Pad's for delivery vehicles) saw the business achieve critical acclaim. Rave reviews in Short Black and Cheap Eats to name a few, were followed by televised appearances on 'Seattle Weekly' and Seattle's Food Lover's Guide. A second Pad restaurant opened in dissertation services Bellevue, and was awarded the prestigious 'Bent Fork award. Dan sold the Pad enterprise in 20XX to analytical rubrics pursue business interests in Thailand.
It was here, while enjoying Bangkok's vibrant street food culture, where the dissertation writing in pakistan, Nudlez concept was born. Essay On Hector From! Dan has a degree in Science from the writing services, University of Washington. In the long term, the employee/income ratio will be approximately $125,000 per employee. Nudlez will run a lean operation with an emphasis on prescribing substances in sales and dissertation writing services in pakistan food production. In years two and three the sales staff will grow from dissertation university phoenix 20 to 110 people, and the production staff will grow from 12 to 277 people. Dissertation Services! Administration will grow as needed. Nudlez will require $1,500,000 from an investor. The funds will be used primarily for the completion of one central and five suburban meal production facilities. The funds will be expected in two rounds, $750,000 to begin operations, and a further $750,000 seven months later (four months after revenues have commenced). The investor's $1,500,000 will be structured as a redeemable note, convertible to 15% shareholding in Nudlez at designer cover ideas the end of year-3.
Alternatively, redemption can be done at this time providing a return of $3,450,000 (2.3 times investment), equating to a compound annual return of approximately 32%. The end of year-3 is good point for possible investor exit, as decisions on product expansion into supermarkets will be made at this time. Funds will be used to initially finish development of the NVU and to establish fit out central premises in Seattle, including IT infrastructure. 5 hubs and dissertation in pakistan 5 sites will initially be established in Seattle, following that expansion into Vancouver/Portland and Spokane will begin. 7.3 INCOME STATEMENT PROJECTIONS. Nudlez will be profitable by letter, the end of its first fiscal year, as shown in the attached financial statements. The business will grow dramatically in years two and three. Break Even Analysis. A break even analysis, assuming zero catering and acquisition stores, reveals less than 100 meals per NVU and dissertation services in pakistan 150 per store need to be sold per trading day to achieve break even.
Number of essay rubrics, average rain days for the 3 cities is dissertation writing services 12 per i like the poetry month. This equates to approximately 40% of the month, which we have conservatively applied, despite its low probability. The conservatism also allows for logistics accidents, staff truancy, etc. to arrive at 13 trading days/month for NVU's. (Exclude catering) Total employment in the expanded Seattle CBD areas is estimated at over 350,000. Nudlez expects to capture less than 2% of dissertation writing in pakistan, this daily market from designer cover letter CBD NVU's and writing stores Seattle and Portland/Vancouver are similar sized markets, with Spokane being approximately half the size. Revenue estimates reflect these ratios. Gen-1 and first Gen-2 NVU, testing and development costs totaling around $75,000 already spent by inventor are not included in essays financials. First Gen-2 NVU will obtain homologation, meaning that the design has blanket pre-approval with local government, reducing time and cost for individual site permits. First 10 NVU's cost $125,000, including homologation. Wastage is in pakistan estimated at 15% across all food related product (validated in cover letter Seattle trials). This will reduce once the 'Nudle-Rac' and associated IT systems are in place Training, Quality Management Systems and services in pakistan Franchise Operations Manuals will be completed effectively for $220,000 by the end of essays, year 1. Copyright © 1998-2017, Khera Communications, Inc.
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