Gun Control Laws Should be Tightened Essay

Nov 02, 2017 Gun Control Laws Should be Tightened Essay, write my paper -

APA Essay Format: Help with Writing Your Essay Paper. An APA , American Psychological Association, style is a standard of writing academic papers in a variety of subjects relevant to the social sciences. This allows to write essays and research papers according to the same generally accepted standard in Laws Should be Tightened, Sociology, Psychology, Education, Political Science, Business and other disciplines. APA Style is room 101 speech, notable for certain requirements related to paper format, font, margins and headings, as well as referencing. We designed this page to guide you through all the peculiarities of formatting a paper in APA style . Learning about APA style will be a very rewarding experience for you, as you will be able to Laws Essay reply it in all of your next writing assignments. Power? Quick Navigation through the APA Essay Format Page: The Fundamentals of an APA Essay Format. What comprises the APA style ? Does it provide requirements only to referencing the sources or to whole essay? In their published guide book, the American Psychological Association, provides APA standards on the following issues: Title page.

Unlike MLA style which doesn#8217;t require a title page, it is a must in APA essay format . It usually contains such standard elements as the title of the paper, the student#8217;s name, teachers or course name, and the due date. The APA title page can be easily distinguished by Gun Control the running header, the page number on room 101 speech the title page and two titles (a short one is followed by the full title). Abstract . Abstract is a single paragraph usually a half page long, and is written on a separate sheet. An abstract summarizing the entire paper has the same copyrights as the whole paper. It should provide the main ideas/results of the paper and Gun Control Laws Should Essay mention the contrastive, methodology used to Gun Control Laws Should be Tightened achieve them. Page format . Page format recommendations in APA style concern page numbers, margins, indentation and spacing.

In-text references . The format of references in endgame, APA format is the foremost subject of student#8217;s concerns. You may pick up citations, quotations and Gun Control Should be Tightened summaries from various sources to support your statements. Define Contrastive? When you use the Should, idea or results that are not yours, they are to be referenced correctly. APA style approves of in-text references. The author and the year of publication should be included within the parenthesis in contrastive, the essay. Page numbers also need to be mentioned when picking up lines from a book. Use of quotations . APA style recommends to put short quotations in quotation marks. If the quotations used exceed the word limit of 40 words, then the writer should indent 5 spaces from the Laws, margin and it should be double spaced. Also, in case of a long quotation, quotation marks should not be used, instead it should be ended with a full stop. Headings . Though it may be not required for an essay, but if you will write a research paper or thesis in quasimodo, APA format you#8217;ll need to structure it.

Headings are used to be Tightened separate and classify paper sections. Thus use different heading styles for headings of sections and subsequent subsections. Reference list . Reference list is a must when you use in-text references, for endgame sparknotes you need to present the full information about the Laws, sources used.The reference list includes all sources used in the essay writing and cited in the paper, and it is arranged according to the alphabetical order by author. It is also of great importance to know how exactly different sources are cited as books, journals, magazines, and web pages are cited in a different way with certain requirements to each type of a source. Room 101 Speech? You may consider how the basic APA requirements are met in APA Essay sample . Laws Essay? APA Essay Template (Cick the Image to Enlarge) When using APA style there are a few standards to keep in mind: double spaced; have all the margins set to one inch; it’s recommended to use the font serif typeface for absolute quote the text and sans serif typeface for any labels; use 10-12 for the font size; always have page numbers; a header with the title of the Gun Control be Tightened, paper should also be used. So, you may either format your essay in APA format yourself or download APA Template in sparknotes, rtf file from P rof E ssays.com . Laws Should Be Tightened? Our expert writers will format your paper for room 101 speech free when you place an order on our website. Your essay will come to you in the APA format with a free bibliography and proper in-text citations. Other free extras that are included are a free outline, free plagiarism report and free title page.

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Learn More. Why choosing us to help? P rof E ssays.com has been providing custom writing services to students for the past eight years. Thousands of students have benefited from our services, achieving excellence in their courses and education. The evident testimony to the excellence of our services and the trust our customers have in Gun Control Laws Should be Tightened Essay, us is that 65% of our customers come back with repeat orders. Your privacy is important to P rof E ssays.com , none of your personal information or details, such as credit card or banking details, will ever be compromised or disclosed to any third party. You are always on the save side with P rof E ssays.com ! Click here to place your order. References are obligatory in endgame sparknotes, a body of the essay if you use some external sources, and especially when you cite them in the APA essay . In-text references are used instead of footnotes in APA format. The sources are indicated by the last name of an author, a year of Gun Control be Tightened publication and a page number (if possible).

In-text references are put in parenthesis (round brackets) within the sentence. Thus the standard in-text reference in APA style will have the following format (Author, year) or (Author, year, page) . But there may be variations: 1. If you are referring to an idea from another work, summarize it findings, or tell about the authors viewpoint you are referring to sparknotes the whole book and should use (Author, year) format. Should Be Tightened? Example : T. E. Lawrence, a British intelligence officer, became regarded as the man who was in charge of the Arab revolt (Thomas, 1924). 2. If you are quoting the source or bring in figures provided in it, you need to include the page number in your in-text reference. Like: (Author, year, page). Example : Lawrence was compared to #8220;a caliph who had stepped out from the pages of #8216;The Arabian nights'#8221; (Thomas, 1924, 16) 3. The Hunchback Dame Quasimodo? If the author or the year of publication is already mentioned in the sentence there is no need to Gun Control Laws Should be Tightened Essay repeat this information in the in-text citation again. Is Here To Stay In America Essay? Examples: With Lawrence in Arabia was published in 1924 (Thomas). Lawrence of Arabia became famous due to the book of Laws Should Essay Lowell Thomas (1924).

4. Multiple authors reference . Room 101 Speech? If the source is written by multiple authors, then you should state last names of all of them, at Gun Control Laws be Tightened Essay, least in the first reference within the essay. Later in the essay, one can use the name of the first author with et al for others. Example: (Morris, Raskin, 1993) first in-text reference. (Morris et al., 1993) subsequent in-text reference. 5. In case of no author . Sparknotes? In some cases the author may be an organization or department, so the Laws Should be Tightened Essay, stating of endgame sparknotes their name will complete the Gun Control, APA essay format . Endgame? Or else the mention of the book will also fulfill the need. Titles of Laws Should be Tightened books in the in-text references should be italicized, titles of articles, chapters, and web pages are in quotation marks. The Hunchback Of Notre Dame Quasimodo? Organization: (ProfEssays.com, 2009) Book: (The correspondence, 1914) The reference list includes all sources used in the course of the essay and cited in the paper. Should Be Tightened? The references should always be organized in alphabetical order . Of Notre Quasimodo? This gives an organized look to the essay. It is also important to know exactly how different sources, such as books, journals, magazines, and web pages should be shown in the reference list. As certain requirements differ for each type of source: Book : Author, A. (Year of publication).

Title of work: SUBTITLE. Location: Publisher. Note: if you have several books by the same author in Gun Control Laws be Tightened Essay, the reference list, you should list them in chronological order. Article : Author, A. (Year). Title of Racism is Here to Stay Essay article. Gun Control Laws Be Tightened Essay? Title of room 101 speech Periodical, volume number (issue number), pages. on-line source : Author, A. A. (Year, Month Date if availiable).

Title of Laws Should be Tightened article. Name of the web-site. Retrieved from absolute, http://www.url/ on Year, Month Date. Readers and writers alike can find headings as a useful tool in writing. Gun Control Laws? Aside from providing order, essay headings can function as an define contrastive, ID – in the sense that it can provide identification on the ideas that are presented below. Headings function as a guide for Laws Should be Tightened your readers, as it will clue them in on your thought flow. The APA style allows five levels of headings when writing. Level one is the first category, while the subsequent levels are provisions for succeeding sub-categories. In the APA essay format it is of the utmost importance to use the titles and headings appropriately. APA makes recommendations on the APA essay headings format, where information on font, punctuation and position are provided, all of which can be viewed at john contribution to atomic, ProfEssays.com : heading level 1: Centered, Boldface, Each Word with a Capital Letter. heading level 2: Left-adjusted, Boldface, Each Word with a Capital Letter. Gun Control Laws Should Essay? heading level 5: Indented, italicized, with a period . Note : in APA format headings #8220; Introduction #8221; and #8220; Conclusion #8221; are not required!

Only the define, parts of the main body should have headings! Title page should be numbered in APA format. A “running header” should be placed in the right corner at the top of the page next to Should Essay the page number. A “running header” in APA style contains a short title of the essay. John Dalton Contribution To Atomic Theory? The essay title should be placed one-third of the Gun Control Should, page down from the top. APA styles requires to write a short title first, put a colon and write the full title underneath. Author#8217;s name is usually several lines underneath.

The name of the course, the name of professor, and the date of submission are printed at the bottom of the endgame, APA title page. APA regulations on abbreviations, tables and figures. Gun Control Laws Essay? You may not need it in your APA essay , but there are APA regulations on abbreviations, tables and figures. Abbreviations can be used later in absolute power, the essay after the full form is be Tightened Essay, mention in endgame sparknotes, the first time. This saves space, but makes a mention of the author, book or organization each time it is used. In APA papers both tables and figures should be presented on separate pages, with the caption at Gun Control Essay, the bottom of the john to atomic theory, page.

What to remember with the APA essay writing format. The main thrust of documenting guidelines is to avoid plagiarism. To do so, several citing outlines have been created by many academics or groups – one of which is the American Psychological Association or the APA. As with other documenting guidelines, the APA generally recommends in-text citations. These types are enclosed in Should, a parenthesis and placed immediately after the borrowed thought or idea.

When beginning with the APA essay writing format , P rof E ssays.com suggests that the student should always remember to take note of the reference material’s author(s), publication house, publication year and define contrastive the page number where the idea was taken from. Gun Control Laws Should Be Tightened? APA essay format may sound tricky and troublesome to some, but is of notre quasimodo, actually an easy way to organize a written work. The APA Essay Format is entirely designed to be Tightened Essay structure the flow of thoughts throughout the essay. It is said that the citing is complicated, but only has a few rules and guidelines that need to be followed. The format can be adapted depending on the nature and absolute the topic of a custom essay. It particularly helps in cases where many writers are involved in the writing process at the same time, giving symmetry and a logical presentation without a miss to the readers. When writing an essay, citing your reference(s) is considered a critical part of the Gun Control Should be Tightened, writing process. This fundamental element of writing helps students avoid plagiarism. In an dalton, academe, plagiarism is a very serious offense that can sometimes lead to expulsion. Laws Be Tightened? The American Psychological Association or the APA style is the formatting guide that is used when writing a paper on the Social Sciences.

As with all documenting styles, the the hunchback quasimodo, APA has recommendations on essay format, headings and citing references. Information on be Tightened Essay formatting your APA essay now easier because there are a number of sparknotes on-line sites, such as P rof E ssays.com , which can provide you with key points in documentation. Anyone and any company can claim to offer its essay help, but not every company can offer high-quality and result-oriented essay help. The appropriate APA format for your manuscript. Many professors recommend strict compliance with manuscript formatting guidelines when writing any type of academic paper. Gun Control Should Essay? Following the format not only ensures organization for define each essay written by an individual student, but it also provides uniformity in appearance for all students’ academic papers. In writing a paper regarding the social sciences, APA makes several recommendations and guidelines on the format of essays . Though these elements may seem trivialon the use of the title page, the format of headings, page numbers, margins, indentation and spacing.

Though these elements may seem trivial, P rof E ssays.com believes that compliance remains a must. College essay format: writing effectively in the collegiate level. What differentiates a college essay from Gun Control Essay, a high school or a junior high school essay is the level of define professionalism it exudes. As more complex topics are discussed, the need for Laws Should Essay further research becomes a necessity. Another important factor to consider is the format of the essay. APA college essay format requires thorough referencing and citing as well as structuring of the essay . While college academic papers follow the quote, basic essay format of introduction, body and conclusion, it is important to remember that the APA college essay format must be more encompassing and Gun Control Should be Tightened more extensive. The APA college essay format differs from simple APA style so we provide specific essay help in this type of writing.

P rof E ssays.com believes that many academes actually require a five paragraph essay on the college level, where the body consumes 60% of the whole essay. The Hunchback Of Notre Dame? P rof E ssays.com is professional and reliable essay writing service which provides 24/7 customer essay help. Write Effectively With the Right Essay Format. A standard essay format provides order to any academic paper. Though this is merely a superficial element of Should be Tightened Essay essay writing, its importance is undeniable, as many established documenting styles provide recommendations for the format of any academic paper. The American Psychological Association, APA, also has suggestions for power your APA college essay format .

Nov 02, 2017 Gun Control Laws Should be Tightened Essay, pay for exclusive essay -

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Anabolic steroid , Carl Lewis , Cheating 1076 Words | 3 Pages. you ever be interested in the topic of war? Fallen Angels is a book about just that, the Vietnam War to of notre dame, be exact. The main character’s name is . Perry, a soldier who was drafted into the military in the 1960’s. In 2002, Fallen Angels was banned in Gun Control Laws some middle schools “for offensive language, racism, violence, and being unsuited to age group.” People think the book has too much profanity and drug use. Fallen Angels should not be banned because it tells young people the history of America.

It gives. Angel , Army , High school 991 Words | 3 Pages. ? bcom/215 Persuasive Essay At least 11% of quote women smoke during pregnancy. Many of these . women smoke without knowing the significant negative effects smoking has on their babies before, during, and after the pregnancy. When a woman smokes during pregnancy, she exposes her baby to dangerous chemicals such as tar, carbon monoxide, nicotine, and other poisonous chemicals that travels through the bloodstream and goes directly to Gun Control Laws Should Essay, the baby. Nicotine restricts the oxygen. Acute respiratory distress syndrome , Breastfeeding , Childbirth 788 Words | 3 Pages. ? Sydia Smith December 17, 2013 ENGL 101 Persuasive Essay Assignment: For generations people . of Racism is Here color have been put into a system caste of injustice.

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Persuasive Essay Laziness in Laws Should be Tightened Essay America By Greg Hill Comp 2 America today is a very lazy place; people have become too . Power! accustomed to their ways and will not even consider what it is doing to them. Everyone constantly praises our wonderful technology; what they don’t realize is what our tech has done to us. As we depend more and more on gadgets to do things for us we lose ourselves to apathy and our personal interests completely forgetting the Gun Control Laws Should Essay, common good. When it becomes so easy to Racism to Stay in America Essay, satisfy our. Adipose tissue , Body mass index , Body shape 1046 Words | 3 Pages. ?Hannah Gillespie Mr. Shoupe Period 7 Persuasive Essay 6 October 2014 Don’t Protect Marriage, Expand It Everyone has rights.

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more than one test if the animal pass the conditions to be experimented on. God loved the be Tightened, birds and invented trees. Man loved the birds and invented . cages. Quote! - Jacques Deval, Afin de vivre bel et bien Many scientists say if animal testing is banned how will medical research continue? Are we willing to sacrifice millions of sick and Gun Control Laws Essay, dying loved humans, who have families and complex emotions, for the sake of a bunch of mice who will only john dalton contribution to atomic theory, live a year or two anyway? However, the answer to Laws be Tightened, this question. Animal rights , Animal testing , Mammal 889 Words | 3 Pages.

read a persuasive essay and to atomic, then write one of your own. Gun Control Should Essay! The essay you write will use persuasive . techniques and evidence to argue for adopting a good habit. You will evaluate a peer’s work and contribution to atomic theory, get peer feedback on your own essay using the provided evaluation rubric. Be Tightened Essay! __________________________________________________________________________ ! Directions and Analysis Task 1: Persuasive Writing Using Evidence Forming good study habits is a great way to improve your grades. Read this persuasive essay. Eating , Essay , Habit 1003 Words | 4 Pages. ? Essay 3, Rough Draft March 26, 2012 Should voters be required to is Here to Stay in America Essay, show a photo ID in Gun Control Essay order to vote? The laws that require voters to show . identifications have been around since 1970, but they are becoming stricter in today’s polls. In the past, all that was required was a voter registration card, but now the new laws are requiring voters to dame quasimodo, show picture IDs.

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Brooklyn Museum of Art. New York. 1862. Patterson, Lillie. “Martin Luther King Jr. and the Freedom . Movement”. New York. Laws Be Tightened! John A. Scott. 28 September 1993. Reprint. Trachtenberg, Jeffery A. “Why Book Industry See the World Split Still by Race”. Lili.org. wall street Journal-Eastern Edition, 6 December 2006, Vol.

248 issue 133, p Al-All, 2p. Web. 3 April. 2013 Tribune, McCarthy.” Don’t Rewrite Literature”. Lili.org. philadelphio inquirer, The (PA).

African American , American Civil War , Black people 2056 Words | 5 Pages. Final Persuasive Paper Imagine a country where a woman is depicted as only being used for sex and Racism to Stay, a as slave for their husband. Laws Be Tightened Essay! Imagine a . country in which women are denied an education. Imagine a country where women are sent to jail, beaten, and left to die a long painful death for disobeying an define contrastive order. Gun Control Laws Should Essay! There happens to be a country in sparknotes which all this is true. Women are being persecuted physically, psychologically, and sexually on a routine and daily basis. The government in Afghanistan is likely. Afghanistan , Hamid Karzai , Human rights 2232 Words | 6 Pages. to college would definitely not be affordable for Laws, low-income families and define, possibly for the average paying citizen.

Making those wanting to attend college . unable because it is Gun Control Laws Should Essay, way too expensive. John Dalton Contribution Theory! Not only is the course credit expensive but also the books and ____are overpriced. Sarah Miller, a research analyst at ConvergEx Group has stated that “students shelled out an average of $73,500 in tuition payments over last four years. And when measured against the $84,500 earned by those who entered the. Academic degree , College , Education 1334 Words | 4 Pages. I’ve super sized this paragraph to mimic what McDonalds has been doing to the world’s waistbands for the past 60 odd years. There is one story that I . would like to Gun Control Laws Should be Tightened, mention if only for the sole purpose of quasimodo showing a little objectivity in this essay . Somewhere out in this world there lives a “super man” that has eaten more than 12,000 Big Mac hamburgers in Should Essay his life.

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Alice’s Adventure in the Wonderland The banned book I picked Alice in the Wonderland by Lewis Carroll. In Alice’s . Racism Is Here Essay! Adventure in Gun Control Should be Tightened Wonderland, Alice goes into the world puzzles that seem to have no clear solutions, which copies the ways that life annoys expectations. Endgame Sparknotes! Alice thinks that the solutions she accidents will make a certain kind of logic, but they repeatedly frustrate her control to figure out Wonderland. Alice goes to understand the Gun Control Should Essay, Queen’s silly croquet game but to not reward. In. Alice's Adventures in Wonderland , Cheshire Cat , Christ Church, Oxford 559 Words | 2 Pages. ------------------------------------------------- How to Write a Persuasive (Argumentative) Essay Rating: 5.0 A . persuasive essay is a form of academic writing that is built around a certain argument. Persuasive essays are thus also called argumentative essays . In this type of contribution theory academic paper, the Gun Control Laws be Tightened, writer aims to persuade the endgame sparknotes, reader to accept the Gun Control Should Essay, writer’s point of view, using certain persuasive techniques to prove the point and endgame sparknotes, convince the reader.

Therefore, a successful persuasive essay requires following certain rules. Argument , Essay , Essays 2196 Words | 6 Pages. Does the freedom to own a firearm make you seem more secure? Or does the Laws Should be Tightened Essay, thought of your neighbors owning firearms make you feel less safe? Gun control is absolute power quote, a . subject that has passionate supporters and detractors alike. In the following essay , I will give you some facts that support the Pro Second Amendment argument; in Gun Control Laws Should be Tightened hopes to gain your support in efforts to prevent the government from infringing on our constitutional rights. A well regulated Militia, being necessary to the security of a free.

Crime , Criminology , Firearm 825 Words | 3 Pages. Persuasive Essay w/o Visual Prompt: For this assignment, you are required to endgame, write an argumentative/ persuasive . essay . Gun Control Laws Should! Topic: Think about something that is wrong (as you see it) in America or another country (i.e. your native country). Whatever you choose to write about, you have to make sure that you offer solutions to the problem. This is not a definition essay where you will just define and explain your problem; you have to argue (persuasively) why the problem exists and how the problem. Argument , Argument map , Counterargument 672 Words | 3 Pages.

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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 Gun Control Should be Tightened integers in the number field, the ideals and units in to Stay Essay the ring of integers, the 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. Be Tightened Essay! . . . . . . . . . . . . . . . . . . . . . . . Of Notre Quasimodo! . . . . . . . . . . . . . Gun Control Be Tightened Essay! . . 5 Prerequisites . . . Contribution To Atomic Theory! . . . . . . . . . . . . Laws Should Be Tightened Essay! . . . . . . . Room 101 Speech! . . . . . . . . . Be Tightened Essay! . . . . . . 5 Acknowledgements . . . . . . . . . . . . . . . . . Endgame Sparknotes! . . . . . Gun Control Laws Essay! . . . . . . . . . . . 5 Introduction . . . . . Dalton To Atomic Theory! . . . . . . . . . . Gun Control Be Tightened! . . . . . . . John Dalton To Atomic! . . . . . . . Gun Control Laws Should Be Tightened Essay! . . . . . . . . 1 Exercises . . . . . . . . . . . . . Define! . . . Should Be Tightened Essay! . . . . . . . . . . . . . Define! . . . . . . . . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . Gun Control Be Tightened! . Endgame Sparknotes! . . . . . . . . . Gun Control Laws Be Tightened! . . . . Quote! . . . . . . . . . . . Gun Control Laws! . . Absolute! . . . . . 7 Ideals in products of rings . . . Laws Essay! . . . . . . The Hunchback Dame Quasimodo! . . . . . . . . . . . Should Essay! . . . . . . . . . Dame Quasimodo! . 8 Noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Noetherian modules . . . . . . . . . . . . . . . . Gun Control Be Tightened Essay! . . . . . . . . . . . . . . . . . 10 Local rings . . To Stay Essay! . . Gun Control! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Rings of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Chinese remainder theorem . . Dalton Contribution! . . . Gun Control Should! . . . . . . . . . . . . . . . . . . . . . 12 Review of tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. 2 Rings of Integers 19 First proof that the integral elements form a ring . . . . . . . . . . . . . . . . . . 19 Dedekind’s proof that the integral elements form a ring . Endgame Sparknotes! . . . . . . . Should Essay! . . . . . . Quasimodo! 20 Integral elements . . . . . . . . . . . . Should Be Tightened Essay! . . . . . John Dalton Contribution! . . Gun Control Should Be Tightened! . . . . . . . . . Absolute Power! . . . . . . Laws Be Tightened! 22 Review of bases of A-modules . . Is Here To Stay In America! . . . . . . . . . . . . . Gun Control Should! . . . . . . . Quote! . . . . . 25 Review of norms and traces . . . . . . Gun Control Laws Should Essay! . . . . . . . . . . . . . . . . . . . . . . . 25 Review of bilinear forms . . . . . . . . . . . . . . . Contrastive! . Gun Control Laws! . . . . . . . . . Power Quote! . . . . . 26 Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . Be Tightened Essay! . . . . . . . . 27 Rings of integers are finitely generated . . . . . . . . . . . . . . . . . . . . . . . 29 Finding the ring of integers . . . . . . Absolute Power! . . . . . . . . . . . Laws Should Be Tightened Essay! . . Racism Is Here In America Essay! . Gun Control Be Tightened! . . . . . . . . . 31 Algorithms for finding the ring of integers . Power Quote! . . . . . . . . . . . . . . . . . . . . 34 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gun Control Laws! . . Sparknotes! . . . . 38.

3 Dedekind Domains; Factorization 40 Discrete valuation rings . . . . Laws Should Essay! . . . . . . . . . . . . . . . . . . . . Racism In America! . . Should Be Tightened Essay! . . . . . 40 Dedekind domains . . . . . . . Define! . . . . . . . . . . . . . . . . . . . . . . Gun Control Be Tightened! . Define Contrastive! . . . 42 Unique factorization of Laws Should be Tightened ideals . John Contribution To Atomic Theory! . . . . . . . . . Gun Control Should! . . . . Racism Is Here To Stay In America! . . . . . . . . . . Gun Control Laws Should! . Endgame! . . . 43 The ideal class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gun Control Laws Be Tightened! . . 46 Discrete valuations . . Contribution To Atomic Theory! . . . . . . . . . . . . . . . . . . . . . . . . . . . Gun Control! . . . Define! . 49 Integral closures of Dedekind domains . . . . . . . Be Tightened Essay! . . . . . . . . . . . . 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Laws Should! . . . . . . . . . . . . . . . . . . Racism Is Here To Stay In America Essay! . . Should Be Tightened! . . . . . . . . . 63 Statement of the main theorem and its consequences . . . . . . . . Room 101 Speech! . . . Gun Control Laws Should Be Tightened Essay! . Racism Is Here To Stay Essay! . . . . Gun Control Laws Should Be Tightened! 65 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Some calculus . . . . . . . . . . . . Absolute Power! . . . . . Gun Control Should! . . . . . . . . . . . . . . . . . . . 73 Finiteness of the class number . . . . . . . . . . . . Contrastive! . . . . . . . . . . Gun Control Should! . . . . . 75 Binary quadratic forms . . . . . . . . Room 101 Speech! . . . . . . . . . . . . . . . Laws! . . Contrastive! . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78. 5 The Unit Theorem 80 Statement of the theorem . . . . . . . . . . . . . . . . Laws Should! . . . . . . . . . Room 101 Speech! . . . . . 80 Proof that UK is finitely generated . . . . . . . . . . . . . . . . . Gun Control Laws Should! . . . . . . . . 82 Computation of the rank . . . . . . . . . The Hunchback Of Notre! . . . . . Be Tightened Essay! . . . . . . . . . . Define Contrastive! . . . . . . 83 S -units . . . . . . . . . Gun Control Should! . . . . . Quote! . . . Gun Control Essay! . . . . Quote! . . . . . . . . . . . . . . . . . . . 85 Example: CM fields . . . . . . . . . . . . . . . Gun Control Laws Should Essay! . . . . . Absolute Power Quote! . . . . . . . . . . . . . 86 Example: real quadratic fields . . . . . . . . . . . . . . . Gun Control Should Essay! . Racism Is Here To Stay In America! . . . . . . . . . . . 86 Example: cubic fields with negative discriminant . . . . . . . . . . . . . . . . . 87 Finding .K/ . . . . Gun Control Should Be Tightened Essay! . . . . . . . . . Absolute Power Quote! . . . . . . . . . . . . Gun Control Should Essay! . . . . . . . . . . . 89 Finding a system of fundamental units . . Room 101 Speech! . . . . . . . Essay! . . . . . . . . . . . . . . 89 Regulators . . . . . . . . . . . . . Room 101 Speech! . . Laws Should Be Tightened Essay! . . . . . . . . . . . . . . . . . . . . . . . Endgame Sparknotes! 89 Exercises . . Laws Be Tightened! . . . . . Absolute! . . . . Laws Should Be Tightened! . . . . . . . . . . . . . . . . . . . . . . . . . . . 90. Power Quote! 6 Cyclotomic Extensions; Fermat’s Last Theorem. 91 The basic results . . . . . . . . . . . . . . Should Be Tightened! . . . . . . . . . . . . . . . . . . . Room 101 Speech! . . 91 Class numbers of cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . 97 Units in cyclotomic fields . . . Gun Control Essay! . . . . . . . . . . . . . . . . Racism To Stay! . . . . . . . . . Gun Control Laws Should Essay! . . 97 The first case of Fermat’s last theorem for regular primes . . . Room 101 Speech! . . . . . . . . . Laws Should Essay! . 98 Exercises . . . . . Dame Quasimodo! . Laws Essay! . . Of Notre Quasimodo! . . . . Gun Control! . . . . . . . . . . . . . . . . . . . . . The Hunchback Dame! . . . . Gun Control Should Essay! . 100. 7 Valuations; Local Fields 101 Valuations . . . . . . . In America! . . . . . . . . . . . . . . . . . . . . . Gun Control Should! . . . . . . . . . . 101 Nonarchimedean valuations . . The Hunchback Of Notre Quasimodo! . . . Laws Should! . . . . . . . . Room 101 Speech! . Should! . Absolute Power Quote! . . . Should Be Tightened! . . . . . . . . . . . 102 Equivalent valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Properties of discrete valuations . . . . . . . . . . . Dame Quasimodo! . . . . . . . . . . . . . . . 105 Complete list of valuations for the rational numbers . . . Gun Control Laws Should Be Tightened! . . . . . . . . . . . . . 105 The primes of a number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 The weak approximation theorem . . . . Power! . . . . . . . . . . . . . . . . . . . . . 109 Completions . . Gun Control Should Be Tightened Essay! . . . . . . . . . Of Notre! . . . . . . . . . . . . . . . . . Gun Control! . . John To Atomic! . . . . . . . 110 Completions in the nonarchimedean case . . . . . . . . . . . . . . . . . . . . . . 111 Newton’s lemma . . Gun Control Laws Should Be Tightened! . . . . . . . . . . . . . . . . Absolute Power Quote! . . . . . . . . . . . . . . Laws Be Tightened! . Essay! . 115 Extensions of nonarchimedean valuations . . . . . . . . . . . . . . . . . . . . . 118.

Newton’s polygon . . . . . . . . . . . . . . Laws Essay! . . . . . . To Stay In America Essay! . . . . . . . . . . . . . . 120 Locally compact fields . . . Gun Control Laws Should Essay! . . . . . . . . . . . . . . . . . Define! . . . . . . . . . . . 122 Unramified extensions of a local field . Should Essay! . . . . . . . . . . . . . . . . . . . . . . 123 Totally ramified extensions of K . . . Absolute Quote! . . . . . . . . . . . . . . . . . . . . . . . 125 Ramification groups . . . . . . . . . . . . . . . . . . . . . Gun Control Laws Essay! . Dalton Contribution To Atomic! . . . . . . . . . . . 126 Krasner’s lemma and applications . . . . . Gun Control Laws Should! . . The Hunchback Dame! . . . . . Gun Control Should Essay! . . . . . . . Define Contrastive! . . . . . . 127 Exercises . . . Gun Control Laws Be Tightened Essay! . . . . . . . . . . . . . . . . . Room 101 Speech! . . . . . Gun Control Should! . . . . . Contrastive! . . . Gun Control Laws Be Tightened Essay! . . . . . 129. 8 Global Fields 131 Extending valuations . Contribution Theory! . . . . . . . . . . . . . . . . . . . Laws Should Essay! . . . . . . . . . . . . 131 The product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Room 101 Speech! . Laws Should Be Tightened Essay! 133 Decomposition groups . . . . . . . . . . . . Dalton Contribution To Atomic Theory! . . . . . . . . . . . . . . . . . . . 135 The Frobenius element . . . . . . . . . . . . . . . Laws Essay! . Quote! . . . . . . . . . . Gun Control Essay! . . . . . 137 Examples . . . . . . . . . Absolute! . . . . . . . . . . . . . . . . . . . . . . . Gun Control Should Essay! . . . . . . 139 Computing Galois groups (the hard way) . . . . . Endgame Sparknotes! . . . . . . . . . . . . . . . . . 140 Computing Galois groups (the easy way) . . . . . . . . . . . . . Laws Be Tightened Essay! . . . Room 101 Speech! . . . . . . 141 Applications of the Chebotarev density theorem . . . . . . . . . . . . . . . . . . 146 Exercises . . . . . . . . . . Should Essay! . . . . . . . . . . . . . . . . Define! . . . . . . . . . Gun Control Laws Should Essay! . . . 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 ;. Room 101 Speech! The cardinality of a set S is denoted by jS j (so jS j is the number of elements in Essay 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. Sparknotes! 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 Gun Control be Tightened, 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 power quote finding the Laws, 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 he discovered an early version of the quadratic reciprocity law. LAGRANGE (1736–1813). He found the of notre dame quasimodo, 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 a solution, LEGENDRE (1752–1833). He introduced the Laws Should be Tightened, “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 n. GAUSS (1777–1855). He found the first complete proofs of the quadratic reciprocity law. He studied the Gaussian integers Z?i ? in quote order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is Gun Control Should, closely related to the problem of finding the Racism in America, 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). 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 Laws Should Essay regular primes. HERMITE (1822–1901). He made important contributions to 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 Racism in America Essay, 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 be Tightened, Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion. DEDEKIND (1831–1916). He laid the modern foundations of algebraic number theory by finding the correct definition of the ring of 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 dalton theory, field of p-adic numbers (as the set of infinite sums. 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 his famous problems were on Should be Tightened Essay, number theory, and have also been influential. TAKAGI (1875–1960). He proved the fundamental theorems of abelian class field theory, as conjectured by Weber and Hilbert. John Contribution To Atomic! NOETHER (1882–1935). Together with Artin, she laid the foundations of modern algebra in 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 Gun Control Laws Should be Tightened, Dedekind’s zeta functions. ARTIN (1898–1962). He found the “Artin reciprocity law”, which is the main theorem of class field theory (improvement of absolute power quote 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 Essay central simple algebras over number fields. BRAUER (1901–1977). Defined the Brauer group, and define, contributed to Should Essay, the classification of 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 the hunchback dame 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– ). He proved new results in group cohomology, which allowed him to give an elegant reformulation of class field theory. With Lubin he found an Gun Control Laws Should be Tightened, explicit way of generating abelian extensions of local fields. Is Here! LANGLANDS (1936– ). The Langlands program2 is a vast series of conjectures that, among other things, contains a nonabelian class field theory. Laws Should Be Tightened! 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. Sparknotes! 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 Essay, 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 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 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 room 101 speech, 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 Laws Should be Tightened Essay, 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 endgame sparknotes 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. Laws Should Essay! A nonzero element of an dalton to atomic theory, integral domain A is said to be irreducible if it is not a unit, and can’t be written as a product of two nonunits. For example, a prime element is (obviously) irreducible. A ring A is a unique factorization domain if every nonzero element of Should be Tightened A can be expressed as a product of irreducible elements in essentially one way. Is the ring of power quote integers OK a unique factorization domain? No, not in general! We shall see that each element of Gun Control Essay 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 define contrastive, 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. 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 Gun Control Should be Tightened Essay, a unit. A similar argument shows that 2;3, and 1?. p ?5 are irreducible. Next note that (*) implies that associates have the same norm, and power quote, 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. OK need not be prime. In the above example, 1C p ?5 divides 2 3 but it divides neither 2. nor 3. In fact, in Gun Control 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 contrastive 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; . Laws Be Tightened! ;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. .2;1C p ?5/.2;1?. Sparknotes! Conversely, 2D 6?4 2 .4;2C2. and Gun Control Laws Should be Tightened, 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?. p ?5/, and .3;1?. Room 101 Speech! 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 Gun Control Laws Essay, of the form a .a/ for some a 2 S and Essay, 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 rank 1g: In general, we shall show (unit theorem) that the roots of 1 in Laws be Tightened 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. Define! 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 Gun Control Should, 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 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. 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. 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 Racism is Here, compute the Galois group only Gun Control be Tightened when the degree is very small. By using algebraic number theory one can write down an algorithm to do it for 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 quasimodo, 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 this course, but wasn’t available when these notes were first written. While the 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 Laws be Tightened Essay, 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 how algebraic number theory developed. Edwards 1977 is a history of contribution theory 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 Laws Essay algebraic number theory.

Lemmermeyer 2009, which explains the origins of “ideal numbers”, and 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 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 rings of integers in number fields are true for more general commutative rings, and absolute quote, it is more natural to prove them in that context.1. All rings will be commutative, and have an identity element (i.e., an element 1 such that 1a D a for all a 2 A), and a homomorphism of rings will map the identity element to the identity element. Gun Control Laws Should Be Tightened Essay! 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; . ;?m? denotes the smallest subring of B containing A and the ?i . It consists of 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 john contribution theory, 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 notes A Primer of Commutative Algebra available on be Tightened, 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 it. An ideal a is maximal if and only if A=a is a field. Every proper ideal a of A is contained in 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 b are ideals in A and B respectively, then ab is an ideal in AB , and every ideal in AB is of this form. Contrastive! The prime ideals of AB are the ideals of the 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 Essay, prime if and only if C=c is an integral domain. The map. Dalton Theory! has kernel ab, and hence induces an isomorphism. Now use that a product of rings is an Gun Control, 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 sparknotes 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 Laws be Tightened, D anC1 D . (c) Every nonempty set S of 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 ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and room 101 speech, so an D anC1 D D a. (b) (c): Let a1 2 S . If a1 is not a maximal element of Laws Should be Tightened 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 sparknotes a properly containing a0. This contradicts the Gun Control Essay, 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 endgame symbols is not Noetherian because the chain of ideals. 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. Laws Be Tightened! 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 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 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 endgame, 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. Should Be Tightened! REMARK 1.5 Note that the proposition fails for quasimodo 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 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.

Similar to Gun Control Laws Should Essay, 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 . Dalton! 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 Gun Control be Tightened Essay, ascending chain of submodules of M . Dalton To Atomic! 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 some ideal a in A, and the statement is obvious. Laws Essay! We argue by induction on the minimum number n of generators ofM . 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. Racism To Stay! 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. (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; . ; eng generates M , which contradicts our choice of 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 . Laws Should Essay! 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. The Hunchback Of Notre! 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 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 Should Essay, a nonzero integer, then2 Zd consists of to Stay 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 Gun Control Laws be Tightened which Zp denotes the ring of p-adic integers. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. PROPOSITION 1.11 Consider an 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 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. Clearly .aA/e a because aA a and a is an ideal in S?1A. For the is Here to Stay in America Essay, reverse inclusion, let b 2 a. Gun Control Laws Be Tightened Essay! 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. John Dalton Contribution To Atomic! 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 Laws Should 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 quote, 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. Gun Control Should Be Tightened! EXAMPLE 1.13 (a) If p is 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 general, for t a nonzero element of an integral domain, fprime ideals of Atg $ fprime ideals of 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 power, b in Gun Control Should be Tightened a ring A to be relatively prime if aCbD A. Contrastive! If m1; . Gun Control Should! ;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. Room 101 Speech! Q .mi /. Note that in general, a1 a2 an a1a2 . an; but the two ideals need not be equal. Laws! These remarks suggest the following statement. THEOREM 1.14 Let a1; . ;an be ideals in a ring A, relatively prime in pairs. Then for contrastive any elements x1; . ;xn of Essay A, the congruences. have a simultaneous solution x 2 A; moreover, if x is one solution, then the other solutions are the elements of the john dalton contribution, 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 Gun Control be Tightened Essay, required property. 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. c D a1cCa2c 2 a1 a2. which proves that a1 a2 D a1a2. We complete the proof by induction. This allows us to to Stay in America Essay, assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . The theorem extends to Gun Control Laws Should Essay, 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 room 101 speech, linear in 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 Gun Control Laws Should be Tightened, 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 sparknotes, following two conditions: (a) The mapM N !M ?AN is Essay, 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 dalton contribution, as relations. Gun Control Laws Be Tightened Essay! 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 /. In particular, if V and W are vector spaces over a field k of dimensions m and n respectively, then V ?kW is 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 absolute power, 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. : : : . Gun Control Laws! 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 generated as an A-module by the elements m0?n0, m0 2 M 0, n0 2 N 0. 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 image of ??? contains a set of generators for M 0?AN 0 and so it is endgame sparknotes, equal to it. 2. One can also show that if M 0!M !M 00! 0. is Gun Control Laws be Tightened Essay, 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 A. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of A-modules, and M is said to be a free A-module of is Here rank m. 4Kronecker products of Gun Control Should Essay matrices pre-date tensor products by about 70 years. The Hunchback! 1. 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. 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. 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 A-module, then B?AM has a natural structure of 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 Laws Should be Tightened a B-module N , there is power, a unique B-linear map ?0WB?AM !N such that ?0.1?m/D ?.m/. Thus ? 7! ?0 defines an isomorphism. HomA.M;N /! HomB.B?AM;N/, N a B-module: For example, A?AM DM . If M is 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/. Essay! 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 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 ?. By definition this means that the map g.X/ 7! g.?/ determines an of notre dame quasimodo, 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 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 Laws be Tightened Essay k, and let ? be an arbitrary field extension. Then K?k? is sparknotes, 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 ?i respectively, then. Gun Control Should Essay! 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 Racism is Here to Stay in America Essay 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 ? 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 Gun Control be Tightened algebraic integers, and later by others in polynomial rings. 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 your website.

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Assault , Crime , Crimes 1597 Words | 5 Pages. Centre for Language Study Pre-Sessional Course A 2013 WRITING TASK 4: Assessed Essay WRITING TASK 4 is your assessed essay . and you will need to write an essay of 1200-1400 words. This is the same length as the essay in WRITING TASK 3. WRITING TASK 4 will form 80% of your mark for writing on Laws Should be Tightened Essay, the Pre-Sessional course. Please select a title from the list here: 1. Discuss the impact a particular theory has had on contribution, your area of study in terms of application, use and Gun Control Laws Should be Tightened limitations. 2. Explain the problems. Citation , Essay , Marketing 1150 Words | 4 Pages. All men have crimes , and most of them are hidden. Contrastive. Do you agree with this statement? As a matter of Gun Control Laws Should be Tightened fact, in this modernization and the hunchback of notre dame quasimodo . globalization era, many people go against the laws to commit the immoral activities such as kidnapping, murdering and drug dealing because they are self-centered and they do not care about others' feeling. Why do people commit crime ? Normally, people commit crimes because that is what they want to Gun Control Laws Should, do.

Criminal behaviour is a matter of choices. Today, there are many. Capital punishment , Crime , Criminal justice 878 Words | 3 Pages. Essay #2: What are the roots of the violence/hate crimes today in room 101 speech, our contemporary society? What can we do to reduce them? . Explain.

The world is full of HATE. What is this word? What makes someone HATE someone else enough to kill or harm another human being? Hate crimes are criminal actions intended to harm or intimidate people because of Laws Essay their race, ethnicity, sexual orientation, religion, or other minority group status. They are also referred to as bias crimes . Hate crimes have been going. Assault , Crucifixion of Jesus , Hate crime 925 Words | 3 Pages. United States to positive changes in the advancements of crime fighting.

Technology has also made it possible to communicate with other . countries, aide in the hunchback of notre dame, other nation’s tragedies, predict patterns, and be Tightened even help fight crimes unknown. However, as technology hits the spotlight for improving communication and crime fighting, there is always a downside. Technology can also be used in a negative manner. As technology improves the define advancement of crimes also improves. Technology can also divide countries. Crime , Crime prevention , Criminology 2189 Words | 7 Pages. the rapid growth of globalisation is the increase in Should Essay, international crime . Despite limitations such as different value systems, language . barriers and define enforceability halted due to state sovereignty, domestic and Laws Essay international legal systems attempt to john theory, address growing problems, to a relatively effective degree. The recent Crimes Legislation Amendment Act 2013 (Cth) attempts to impede people trafficking and the Joint Organisation Crime Group of Australian federal and Laws be Tightened state agencies have endeavoured to. Crime , Extradition , Human rights 1557 Words | 5 Pages.

Organized Crime Essay - Ciminology. Organized Crime Organized crime has been around for many years. Since the first definition was . written organized crime has grown into multiple different meanings. According to the text, Introduction to Criminology, organized crime refers to in America, “organizations that use violence, provide illegal services, and have immunity of operation.” (Hagan) Organized crime is also used throughout the world within gangs and secret organizations. Laws Be Tightened. There are multiple types and definitions. Crime , Crimes , Criminology 2346 Words | 6 Pages. Through the film L.A. Confidential by Curtis Hansen and the novel The Life and Crimes of Harry Lavender by Marele Day, the the hunchback composers manipulate . the expectations and conventions of crime genre to generate their own representations of society. These crafted views of crime in society position the audience to be Tightened, view certain aspects of society differently, and to question the boundaries and norms currently set up. The audiences of both texts are encouraged to take a position on the truth behind facades. Crime , Crime fiction , Detective fiction 1347 Words | 4 Pages.

development of organized crime and organized crime groups will be discussed. “The corrupt political machines may directly affect . social disorganization and the criminal activity of organized crime groups” (Wilson, 2012, p.01). John Dalton. It is important to note that social disorganization is contributed by Gun Control many factors. Social disorganization Andres Rengifo (2009) describes social disorganization as “a theoretical perspective that explains ecological differences in levels of crime based on structural and absolute power cultural. Corrections , Crime , Criminal justice 895 Words | 3 Pages. 115), so it’s easy to Laws Should be Tightened Essay, use and set up, install and maintain. The monitors are placed in certain areas of the store or strategic places that meet their needs. . The good thing about these types of cameras is absolute power quote, they not only alert to shoplifters but other crimes as well (pg.

115). Because they are also set up outside in parking lots and tape continuously they can get license plates or pictures of faces in case of kidnaps or murders, and muggings as well if there in range of the cameras. Security. Crimes , Police , Security 750 Words | 3 Pages. Link of Should be Tightened Genetics and Criminal Behavior AIU Online Abstract In this essay the quote link between criminal behavior and Gun Control Laws Should genetics . will be discussed. Also in endgame, this paper the discussion on Gun Control Laws, how crime , twin studies, adoption studies, and testosterone studies are linked to genetics will be discussed. This essay will show some of the strengths and weaknesses of the evidence that is define contrastive, linked to genetics and crime . Laws Should Be Tightened. I will also be giving my feelings on whether or not I agree with police departments. Crime , Criminal justice , Criminal law 1494 Words | 4 Pages. going to make poor choices. Room 101 Speech. We all hope the Gun Control Should be Tightened impact of those choices will be minor, provide opportunities to quote, learn and Should be Tightened Essay leave no lingering consequences for dalton to atomic theory . your children or others. When a child who has been taught normative behavior chooses to Gun Control, commit a crime , then he or she should be held responsible.

The only exceptions are when a parent has become aware of a situation that might have been prevented had they known about it and failed to intervene or when they provide the tools that lead to criminal acts. Childhood , Crime , Criminology 1006 Words | 3 Pages. Argumentative Essay Social responsibility is an ideal topic for debate; there have been mixed results for companies and individuals who have . Absolute Power Quote. pursued social responsibility. There is also the Laws Should question of whether social responsibility should be motivated by a perceived benefit.This type of essay is based on philosophical theories on the necessity of social responsibility backed up with facts about previous social responsibility efforts. For example, an essay could be about how giving support to disaster. Essay , Essays , Qualitative research 555 Words | 3 Pages. Bankers Adda How to write Essay in endgame, SBI PO Exam? Dear readers, as you know that SBI PO 2014 Paper will also contain a Descriptive Test of be Tightened . John Contribution To Atomic Theory. 50 marks (1 hour duration), which will consist of English Language Comprehension, Short Precis, Letter Writing Essay ). Gun Control Laws Be Tightened Essay. So, here we are presenting you How to dame, write Essay ? and Gun Control Laws few points to remember while writing an essay in of notre, the exam, which will be important for upcoming SBI PO exam.

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Abstraction , Essay , Fiction 876 Words | 3 Pages. BM 6105 Assignment BM 6105 Managing Change Assignment (3000 words) Essay Due on Gun Control Laws be Tightened Essay, Monday 14th of quasimodo January 2013 You are required to write an . essay supported with reference to the academic literature that answers the following question: • You have recently been appointed to your first management post following graduation. Laws Should. You are keenly aware that as part of your management role you will be responsible for managing change and absolute power anticipate drawing on your BM 6105 studies to help you achieve success. Essay , Management , Organization 690 Words | 3 Pages. Semester 1, 2013 Assessment Task 2:Critical Essay IDEAS in MANAGEMENT Writing instructions and Marking Rubric This assessment task is . an ESSAY . The RMIT College of Business requires you to use a particular style of essay writing which involves both the way the essay is structured and the way that you acknowledge other people’s ideas used in your work. The structuring of an essay is very clearly described in the RMIT Study and Learning Centre Essay Writing Skills Online Tutorial available. Article , Citation , Critical thinking 807 Words | 3 Pages. Computer crime is an ongoing issue that we continuously see, and a major problem with this is that most people do not realize the harm that . computer crime can cause. Computer crime is also a niche that continues to Laws be Tightened Essay, grow.

This is based on the fact that there are many different perceptions about what computer crime is, and the harmful effects it can have. Because there is Racism is Here to Stay, a gap between traditional views of Gun Control Should Essay what cyber crime is and the actual realities of these crimes , it is a criminal offense that will. Average Joe , Computer crime , Computer crimes 1141 Words | 4 Pages. their motivation was will determine if their act was moral or not. By betraying his own kin Momutu decided to give up his humanity, while on the other . Racism Is Here To Stay In America Essay. hand, Amoo gave up his own freedom to protect his family. Testing with success series The Essay Exam Organization and neatness have merit Before writing out the exam: Write down their key words, listings, etc, as they are fresh in your mind. Gun Control Laws Be Tightened Essay. Otherwise these ideas may be blocked (or be unavailable) when the time comes to write. African slave trade , Answer , Atlantic slave trade 857 Words | 4 Pages. such strategies as scanning, skimming, main ideas, contextual clues and inferences. Learning Outcomes: Upon completion of this subject, student will . be able to: 1. write summaries as well as process, comparison-contrast and cause-effect essays 2. apply basic grammatical concepts in writing 3. answer questions based on academic texts 4. Of Notre Dame. give oral presentations Textbook: 1. Daise, D., Norloff, C., and Carne, P., (2011). Q: Skills for Success 4 : Reading and Writing Oxford University.

Cambridge , Essay , Latin 401 Words | 3 Pages. Name: Andrew Gordon Subject: Exp201, Professor Morales Summary Assignment Forerunners Sei Shonagon: Hateful Things Essay question: . Gun Control Laws Should Be Tightened. Consider Sei Shonagon’s definition of the word ‘hate’ in the Heian era. Is an opinion considered hostile if it focuses on honesty and is Here Essay transparency of a situation? “Hateful Things” is an opinionated extract from the book “Pillow Talk” written by Sei Shonagon. Pillow Talk is a collection of the lists, desires, poetry and judgmental conversations by Shonagon. Sei.

Essay , Heian period , Japan 1371 Words | 4 Pages. trying to know everyone in the class. Laws Be Tightened. After that here comes the big turn, homework and writing essays . I have never written an define contrastive, . Gun Control Laws Should Be Tightened Essay. essay in my life while I was living in India. It was really hard for room 101 speech me to write down our thought on the given assignment. Gun Control Laws Be Tightened Essay. My English is dame, not good and grammar is worse. I was really afraid of writing, I didn’t want to tell anyone. I missed my first essay due date and even second essay due date. Gun Control. After some days Mrs. Danielo called me in her office during office hours. She told. Academic term , College , Essay 1524 Words | 4 Pages.

symbols which can be understood and is Here to Stay manipulated by someone who is culturally literate. Second, being literate can mean having knowledge or competence. For . example, we speak of people being computer literate or politically literate. For your first essay , try to focus on Laws Should, a moment or a period in your life when you realized the significance of sparknotes being literate in Gun Control Laws Should be Tightened, this fashion. Did you have trouble using a computer to register for classes? Did you fit into a subculture because you learned to room 101 speech, speak its. Essay , Knowledge , Literacy 1120 Words | 4 Pages. Is Compassion Really a Crime? Analysis of A Crime of Compassion by Barbara Huttmann. Is Compassion Really a Crime ? Imagine standing in a hospital room watching a man suffer in agonizing pain day after day and not being able to Laws, . do anything about it.

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What Is Crime ? by Lawrence M. Friedman For Your Journal How would you answer the question “What is crime ?” For you, what makes . Is Here To Stay In America Essay. some acts criminal and others not? Explain. There is no real answer to the question, What is crime ? There are popular ideas about crime : crime is bad behavior, antisocial behavior, blameworthy acts, and the like. Gun Control Laws Essay. But in of notre dame quasimodo, a very basic sense, crime is a legal concept: what makes some conduct criminal, and Laws Should be Tightened other conduct not, is the fact that some, but not others, are. Crime , Crimes , Criminal justice 929 Words | 4 Pages. theories of youth crime This essay will discuss the understanding of the absolute power sociological and Laws Should psychological factors of youth . crime . Room 101 Speech. It will be agreeing and disagreeing in the above statement Youth crime is also known as juvenile delinquency, juvenile delinquency refers to Laws Essay, criminal acts performed by juveniles. Most legal systems prescribe specific procedures for dealing with juveniles, such as juvenile detention centres. Racism Is Here Essay. There are a multitude of different theories on the cause of Gun Control Laws be Tightened Essay crime , most if not. Conduct disorder , Crime , Criminal justice 1959 Words | 6 Pages. Four types of essays exist including: narration, description, exposition, and argument.

Each type has a unique purpose: some tell a story, some . are descriptive and others prevent viewpoints. One of the best ways to better understand each type of essay is to room 101 speech, review examples. Types of Essays Narrative Narration is telling a story from a certain viewpoint, and there is usually a reason for the telling. All narrative essays will have characters, setting, climax, and most importantly, a plot. Essay , Essays , Exposition 2198 Words | 7 Pages.

? Crime as a complex and contested concept Introduction Crime is a complex and contested concept which will be examined . through different definitions to discuss the strengths and weaknesses of collected information about crime and Gun Control Laws Should be Tightened Essay how it affects the absolute quote way society receives information on crime . Should Essay. This essay will discuss crime to determine why it is contested and endgame received in such a broad category and Essay how interpretations and misrepresentations of crime can make and change society’s views on crime . . Crime , Criminal justice , Criminology 1561 Words | 4 Pages. Essays are generally scholarly pieces of writing written from an author's personal point of of notre view, but the definition is vague, overlapping with . those of an article, a pamphlet and a short story. Essays can consist of a number of elements, including: literary criticism, political manifestos, learned arguments, observations of daily life, recollections, and reflections of the author. Gun Control Laws. Almost all modern essays are written in prose, but works in verse have been dubbed essays (e.g. Alexander Pope's. Alexander Pope , Essay , Essays 1053 Words | 4 Pages. Essay Writing ( The writer expresses his personal ( The writer presents arguments opinion on a topic. for absolute power quote and . against be Tightened a particular statement. |Introduction |Introduction | | | | |• General statements . 2005 albums , Conclusion , Essay 1121 Words | 7 Pages.

Understanding a Criminal Crime amp; Society SOC 305 Angelique Lees July 16, 2012 Understanding a Criminal Crime . Dalton Contribution To Atomic. is an Should be Tightened Essay, activity that societies around the globe have had to define contrastive, experience from Gun Control Laws Should be Tightened, close to the beginning of manhood. Crime is nothing new to us but the room 101 speech extent of criminal behavior has evolved over Gun Control Laws Should Essay the centuries. We will probably never live in endgame, a world that is Gun Control Laws Should be Tightened Essay, free of crime , but learning more about the criminal aspects in our society may one day make our world a much better place. Corrections , Crime , Crime statistics 2315 Words | 7 Pages. Introduction Fear of Crime in members of dalton contribution theory our society today has been widely researched. For the purpose of this essay , fear of . Gun Control Laws Essay. crime is used in room 101 speech, the context of an individual’s perceived risk of becoming a victim of Laws be Tightened Essay crime . In this essay it is room 101 speech, argued that the Gun Control Laws be Tightened Essay elderly and the youngest members of our society are the quasimodo most fearful of Gun Control Laws Should be Tightened crime and that, of of notre dame quasimodo these age groups the elderly have the Laws Should lowest risk of dalton to atomic becoming victims of crime . Firstly, research shows that fear of Laws Should crime is Racism is Here in America Essay, wide spread and that certain. Ageing , Crime , Criminal justice 1936 Words | 5 Pages. Melanie Requena December 3, 2012 Essay # 5 Stereotyping happens all the time, more than people seem to believe it does. Appearance, . location, and race all deal with how people as well as police authority characterize individuals without truly knowing how that person is. Race is Laws Should be Tightened Essay, one of the in America Essay biggest factors when it comes to criminalization because of Should Essay what people see on the media and from what they are told. It usually is the younger generation that are viewed as criminals not only the hunchback dame quasimodo because.

Crime , Criminology , Gang 861 Words | 3 Pages. ?Introduction This essay will critically examine the strengths and weaknesses of the theory that young people offend because of their . upbringing. The term ‘upbringing’ means the care and teaching received by the child from the parent throughout their childhood. Gun Control Laws Should Be Tightened. There has been extensive research and controversial debate into upbringing being the root cause of youth crime and this essay will examine evidence to support this claim and evidence to dispute it. Although it is quite subjective as to. Abuse , Childhood , Crime 1932 Words | 5 Pages. Writing and Informative Essays Informative. hehehehe i actually only wanted toExamples of Informative Essays The purpose of an informative essay , sometimes called an . expository essay , is to educate on a certain topic. It is not for giving an opinion or convincing someone to do something or change his beliefs. In addition to being informative, it needs to be interesting.

Structure of an Informative Essay The basic structure of an informative essay is very simple. It needs to have a beginning, middle, and end. The beginning needs. Blood , Blood donation , Essay 745 Words | 3 Pages. Sociology Professor McGowan 11/23/12 Crime Crime is an act where someone breaks the law or illegal activities. John Dalton Contribution To Atomic Theory. There . different types of Laws crimes such as vice crimes , street crime and computer crime . Sparknotes. Vice crime is a crime where someone is doing a criminal act such as gambling or drug use. Street crime is Gun Control, something down the lines of rape, robbery, or assault.

Finally is computer crime , which would be hacking on define, the computer for Laws Should money or personal mail. These crimes are different and can affect our society. Computer crime , Crime , Criminology 1327 Words | 4 Pages. To what extent is the media’s portrayal of crime balanced and accurate? The essay will consider how accurately the media . portrays crime , this will entail what types of crime stories they report on, whether or not those stories are biased, the impact they have on the public’s perception of crime and the actual reality of crime in Australia. When we talk about the room 101 speech media we are not just referring to Should be Tightened Essay, one specific example of media, but many different types.

The most common forms of room 101 speech media people talk. Concentration of Gun Control Should be Tightened Essay media ownership , Crime , Criminology 2002 Words | 5 Pages. best define crime ? Discuss. Define Contrastive. The Oxford English dictionary defines crime as “an act punishable by law, as being forbidden by . statute or injurious to the public welfare, an evil act; an offence, a sin, -an act can only be considered a crime when identified as such by law. An act was defined a crime in Gun Control Laws Should Essay, the old testament with the creation of the Ten Commandments. This was when it was literally set into stone that numerous acts became a crime against God, the first rules of the world. Crimes are now defined. Crime , Criminal justice , Criminal law 1932 Words | 5 Pages. Thesis: Writing and absolute Synthesis Essay.

?How to be Tightened Essay, Write a Synthesis Essay ?Synthesis Writing: to combine the ideas of more than one source with your own. ?Key Features of a . Synthesis Report information from the is Here to Stay sources using different phrases and Laws Should Essay sentences; Organize so that readers can immediately see where information from the sources overlap; Make sense of the sources and help the reader understand them in greater depth. ?Preparing to write your Synthesis Essay You should locate two or more sources for synthesizing. Begin , Citation , Essay 862 Words | 4 Pages. Essay Examples Four types of sparknotes essays exist including: narration, description, exposition, and argument. Each type has a unique . Gun Control Laws Should Be Tightened Essay. purpose: some tell a story, some are descriptive and others prevent viewpoints. One of the best ways to better understand each type of essay is to the hunchback quasimodo, review examples. Types of Gun Control Laws be Tightened Essay Essays Narrative Narration is telling a story from a certain viewpoint, and there is usually a reason for absolute quote the telling. All narrative essays will have characters, setting, climax, and most importantly. Essay , Essays , Exposition 1903 Words | 6 Pages. . Improving Crime Prevention Knowledge and Practice Essay. Title Homel, P., 2009.

Improving crime prevention knowledge and practice. Laws Should Essay. Trends issues in crime and room 101 speech criminal justice, . November, Issue 385, pp. 1-6. Topic Existing research helps identify the importance of developing of a national framework for technical support to ensure an effective sustainable crime prevention effort. The author suggests the Laws be Tightened need to consider the establishment a crime prevention technical assistance fund. The Hunchback Of Notre Dame. Context During the late nineties Australian research started to Gun Control Laws be Tightened Essay, recognise.

Crime , Crime prevention , Criminology 1786 Words | 6 Pages. whether an individual’s environment can lead them into crime and to behave in a deviant manor. There are seen to be many factors that can be . perceived to the reasoning behind why a proportion of Racism to Stay in America Essay society chooses to Gun Control Laws be Tightened Essay, turn to crime or behave in a deviant way. Sociologists have been divided by certain theories such as environmental matters Durkheim (1964), biological circumstances Lombroso (1879) and endgame psychological issues Eysneck’s (1977). This essay will explore research and theories to Laws Essay, see whether it is. Crime , Criminology , Marxism 1980 Words | 5 Pages. TEN KEYS TO WRITING THE ESSAY Key 1: Budget your time carefully. You have only a half hour in which to complete your work. You should use . your time more or less as shown below: Reading and contribution theory thinking about the topic 2-3 minutes Planning and taking notes 2-3 minutes Writing the essay 20 minutes Checking the essay 3-5 minutes Key 2: Read the question carefully. Laws Be Tightened Essay. You must write on the topic exactly as it is given, so be sure that you understand it.

If you write about another topic, you. Clause , Conclusion , Dependent clause 1149 Words | 4 Pages. Essay Writing: Writing with Purpose. One-word essay : The one-word essay is quite manageable as it allows you to decide which essay type you are going . to write about. Take the topic “Floods”. If you choose to write an expository essay , you can write about the causes and effects of floods; if you choose to write a narrative essay , you can narrate a story related to floods; and if you choose to john dalton to atomic, write a descriptive essay , you can focus on Gun Control Laws Should be Tightened Essay, describing floods.

Descriptive Essay : WRITING a descriptive essay is perhaps more difficult. Conclusion , Essay , Essays 2047 Words | 7 Pages. 06 August 2013 Juvenile Crime In “Adult Crimes , Adult Time,” published in the Washington Post on March 29, 1998, Linda J. Endgame. . Collier argues about the juvenile laws and her principal argument shows that children should serve the same time as adults when they commit adult crimes . Gun Control. Collier provides some statistics and examples (Jonesboro shooting, Daily City shooting, and her experience with a particular girl) that support her position regarding the information of the article. The diversity of laws among. Conduct disorder , Crime , Criminal justice 943 Words | 3 Pages.

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Bedenk: waar kun je essays aantreffen? In welke media, op welke plekken? Wat doen die essays? Wat is Racism Essay, hun doel, hun functie? Probeer voor jezelf te formuleren wat essays zijn. Deel 2: Op zoek naar de persoonlijke vraag. Wat een essay precies is, is be Tightened Essay, nog altijd voer voor vele discussies.

Is elke willekeurige column en recensie in define een dag- of weekblad er een, hoort een biografie ook tot het genre, en kun je alle schoolopstellen essays noemen? Dat laatste is Gun Control be Tightened Essay, nog niet eens zo’n rare veronderstelling, als je naar het curriculum van Amerikaanse en Engelse scholen kijkt. Essayschrijven wordt daar gedrild: inleiding, these, uitwerking, conclusie. Ook de dikke Van Dale lijkt een Angelsaksische knieval te maken door een essay te definieren als ‘een persoonlijk gekleurde verhandeling over een wetenschappelijk of endgame sparknotes, letterkundig onderwerp’, met daarachter een grote pijl en het woord ‘opstel’. Er zijn echter verschillende redenen om het schoolopstel niet tot de kerngroep der essays toe te laten. Een van de belangrijkste argumenten is Gun Control Should be Tightened, dat een echt essay een persoonlijk onderzoek is, een gedachtenexperiment waarbij je van te voren niet weet waar het je zal leiden. Contrastive? En zelfs als je nergens uitkomt, dan nog is Gun Control Laws Essay, het soms boeiend om op te schrijven hoe je van punt A naar punt G kwam, via F, S, L en – wederom – A. Define Contrastive? Een opstel heeft daarvoor een veel te rigide structuur. Gun Control Be Tightened Essay? Alleen al die verplichte conclusie: dat heeft weinig te maken met het meanderende, onderzoekende karakter dat het echte essay kenmerkt.

De Franse werkwoorden assayer / essaier die ten grondslag liggen aan het genre betekenen dan ook proeven, toetsen of contribution theory, uitproberen. ‘Essayisten zijn vrijbuiters, ze zijn niemand ook maar een zier verplicht,’ concludeert Tanny Dobbelaar dan ook in Essay haar boek Schrijven met Montaigne (besproken in sparknotes Club Schrijven Magazine 9/5). Wat zijn dan wel ‘echte’ essays, welke horen tot de kerngroep? Een naam werd hiervoor al even genoemd, namelijk de grondlegger van het genre, Michel de Montaigne (1533-1592) die zich halverwege zijn leven terugtrok op zijn landgoed om zijn gedachten te ordenen en te scherpen in een nieuwe vorm, die hij de naam essai meegaf. Gun Control Laws Essay? Wie Montaignes drie kloeke delen essays leest (niet te snel, ze zijn voor fijnproevers), ziet al snel wat deze onderscheidt van opstellen, scripties en dissertaties. Bij een Montaigne-essay ga je namelijk uit van je persoonlijke ervaringen, gedachten en emoties – iets wat je in endgame sparknotes (semi)wetenschappelijke werk juist te allen tijde probeert te vermijden. Gun Control Essay? Vrijwel alle goede essays geven dan ook inzicht in het persoonlijke leven of in de persoonlijke gedachten van de auteur. Niet omdat die belangrijk zijn (soms gaat het om hele eenvoudige waarnemingen), maar omdat ze een aanleiding kunnen vormen om algemeen geldende ideeen te onderzoeken. Wanneer kun je zeggen dat iemand een gelukkig leven heeft gehad? Wat betekent het om verbannen te zijn?

Mag een arts fouten maken? Wat is contribution, de waarde van vriendschap? Dat soort kwesties worden door Montaigne en zijn nazaten onderzocht. Uiteindelijk gaat het niet om ‘hoe je ergens over Laws denkt’, maar ‘hoe je ergens over define contrastive denkt’. Gun Control Laws Be Tightened? Ofte wel: het gaat niet om je mening, niet eens om je argumenten, maar om de manier waarop je redeneert, voelt, verbanden legt: de logica, evenzeer als de plotselinge invallen. Als je een goed essay schrijft, geef je de lezer een kijkje in Racism in America je geest. Zoek een vraag, een kwestie, een zorg die al een tijd in Gun Control Laws be Tightened Essay je hoofd rondwarrelt. Absolute Power? Je hoeft er nog geen antwoord op te weten. Laws Should? Liever niet. Of Notre? Probeer je gedachte op te schrijven en kijk of Gun Control be Tightened Essay, je er een algemene, abstracte vraag uit kunt afleiden.

Elke essay begint in room 101 speech wezen met zo’n vraag. Gun Control Should Be Tightened? Kijk vooral naar kwesties waarover je twijfelt: hoe duidelijker het antwoord je voor ogen staat, hoe ongeschikter het onderwerp is Racism to Stay, voor een essay. Formuleer nu je basisvraag. Zoek een vraag, een kwestie, een zorg die al een tijd in Should be Tightened je hoofd rondwarrelt. Is Here To Stay Essay? Je hoeft er nog geen antwoord op te weten. Liever niet.

Probeer je gedachte op te schrijven en kijk of Gun Control Laws Essay, je er een algemene, abstracte vraag uit kunt afleiden. In America Essay? Elke essay begint in wezen met zo’n vraag. Be Tightened Essay? Kijk vooral naar kwesties waarover je twijfelt: hoe duidelijker het antwoord je voor ogen staat, hoe ongeschikter het onderwerp is room 101 speech, voor een essay. Formuleer nu je basisvraag. Een persoonlijke aanleiding, zorg of Laws Should be Tightened, vraag is contrastive, het beginpunt van een essay, maar daar moet het niet bij blijven. Gun Control Laws Should Be Tightened? Met een essay wil je uiteindelijk proberen uit je persoonlijke ervaringen een algemene kwestie aan te snijden en te onderzoeken, zoals Jorge Luis Borges deed in define zijn beroemde mini-essay ‘Argumentum Ornithologicum’ uit het boek De Maker . ‘Ik sluit mijn ogen en zie een zwerm vogels.

Het visioen duurt een seconde of misschien korter; ik weet niet hoeveel vogels ik heb gezien. Should? Was hun aantal bepaald of onbepaald? Deze vraag impliceert die naar het bestaan van God. Als God bestaat, is endgame, het aantal bepaald, want God weet hoeveel vogels ik heb gezien. Gun Control Be Tightened? Als God niet bestaat, is define, het aantal onbepaald, want dan kan niemand de telling verrichten. In dat geval zag ik niet negen, acht, zeven, zes, vijf, vier, drie of be Tightened Essay, twee vogles. Ik zag een aantal tussen de tien en een, dat niet negen, acht, zeven, zes, vijf, enz., is.

Een dergelijk getal is dalton theory, niet voorstelbaar: ergo, God bestaat.’ Slechts tien regels telt het essay, dat begint met de beroemde zin: ‘Ik sluit mijn ogen en zie een zwerm vogels.’ Hoeveel vogels het zijn weet Borges niet, en juist dat intrigeert hem. Als God bestaat, weet hij het juiste aantal en is Gun Control Laws Should, het een bepaald getal. En als God niet bestaat is het een onbepaald getal, want niemand heeft het aantal vogels geteld: het kunnen er drie, vijf, acht of negen zijn. The Hunchback? En dus zag Borges in Laws Should be Tightened Essay die flits een getal dat niet twee, drie, vier, etcetera was, en dat kan niet. En dat bewijst dat God bestaat. Van een eenvoudig fantasiebeeld naar een Godsbewijs in contrastive tien zinnen: dat is Essay, het essay ten voeten uit. Wat ook goed te zien is endgame sparknotes, aan dit voorbeeld is hoe een essay bij de lezer werkt: het biedt geen voordehandliggende conclusie, maar zet aan tot denken, tot redeneren, tot het verzinnen van tegenargumenten. Klopt het wel wat Borges beweert? Hoezo is een onbestaand getal een bewijs dat God bestaat? Kun je het bestaan van God wel beredeneren?

De lezer maakt als het ware zelf het essay af. Bedenk hoe je zelf een lezer over Should Essay je onderwerp kan laten nadenken. John Dalton Contribution Theory? Wat prikkelt hem? Hoe kun je hem laten nadenken? Natuurlijk niet door hem alleen maar te overladen met argumenten en voorbeelden. Gun Control Laws Should Essay? Hoe moet je redeneren, schrijven, om je lezer in absolute power zijn hoofd je essay 'af te maken'? Om een essay goed op te kunnen bouwen, moet je als schrijver eerst vele wegen in be Tightened je hoofd aflopen. Racism In America? Sommige wegen lopen al snel dood, andere zijn te evident: daarvoor hoeft een lezer je artikel niet te lezen. Gun Control Essay? Zoek dus naar de onverwachte sporen, die je een eind verder brengen. Doe dit door jezelf voortdurend vragen te stellen, argumenten en voorbeelden te verzinnen, losjes te associeren. Je materiaal mag je overal vandaan halen, van wetenschappelijke rapporten tot songteksten.

Juist die vele mogelijke dwarsverbanden onderscheiden een essay van een wetenschappelijke tekst. Stel bijvoorbeeld dat je je al een tijdje ergert aan mensen die trein en tram proberen in quote te komen voordat iedereen de trein verlaten heeft – een verschijnsel dat de laatste jaren steeds vaker de kop opsteekt. Laws Essay? Als je rechttoe rechtaan je ergernis zou uitschrijven, zou je geen essay hebben, maar een opinierend stuk, een column. Sparknotes? Je onderzoekt namelijk niets, poneert alleen maar. Dus: zet je ergernis opzij en probeer onder, achter, in, over Gun Control Should be Tightened Essay het probleem te duiken. Sparknotes? Je medepassagiers voor laten gaan is Should Essay, een kwestie van etiquette, maar wat is de functie van etiquette? Kan die functie veranderen?

Ken je nog meer voorbeelden van modern voordringen? Waar doet je die langs elkaar duwende en schuivende massa aan denken, welk beeld komt in to atomic je op? Hoe gaat het in de ons omringende landen? Hebben we eigenlijk al een naam voor de voordringers? En hoe noemen we de mensen die zich laten wegdringen? Probeer in Gun Control Should Essay deze fase zoveel mogelijk ideeen en sporen te verzamelen: het bekritiseren en uitwerken ervan komt later. Ga terug naar je basisvraag of of notre quasimodo, –kwestie uit oefening Zet deze middenop een vel papier. Gun Control Be Tightened Essay? Associeer nu vrijelijk en zet de beelden, woorden, vragen, voorbeelden en argumenten eromheen. Schets met pijlen welke elementen met elkaar te maken hebben.

Doe nu hetzelfde met de beelden, woorden, vragen en voorbeelden: werk ze verder uit door ze op hun beurt weer te omringen met eigen beelden, woorden, vragen en voorbeelden. Contrastive? Hoe ga je hier een essay uitschrijven? Bedenk een route, een structuur, een vorm. Na het vrijelijk associeren begint het bekritiseren van je eigen ideeen: welke zijn te evident, welke sporen lopen dood. Gun Control Laws Essay? Zoek naar de gedachtensporen waarvan je – vaak intuitief – vermoedt dat er veel achter schuilt. In het voorbeeld van de voordringende treinreizigers kun je bijvoorbeeld al snel vermoeden dat de gedachten over john dalton contribution theory etiquette een beetje voor de hand liggen. Laws Should Be Tightened? En een sociologische verkenning van treingedrag bij onze buurlanden is meer iets voor Elsevier. John Contribution? Zoek naar een vager, lastiger spoor, zoals de naamgevingskwestie of Laws, een intrigerend beeld dat spontaan bij je opkwam, toen je over room 101 speech dit onderwerp nadacht. Gun Control Essay? Wat je altijd kunt doen: een paar uur op een groot station doorbrengen en observeren wat er nu precies gebeurt. Nogmaals: niet om je te ergeren (probeer die emotie even aan de kant te schuiven), maar om te weten wat dat nu precies inhoudt, dat voordringen, hoe het eruit ziet, wat de geluiden zijn, de geuren, de kleuren. Het gebruik van je zintuigen is sowieso een belangrijk onderzoeksmiddel bij het schrijven van essays.

Je bent je eigen controlemiddel, je eigen meetinstrument. Sparknotes? Een goede essayist is Laws Should be Tightened Essay, een scepticus, iemand die veel bekritiseert, over weinig een pasklaar oordeel klaar heeft, en alles zoveel mogelijk onderzoekt. Room 101 Speech? De zintuigen spelen daarbij een cruciale rol. Wat betekent vriendschap? Observeer ‘ns een verjaardagsfeestje, probeer in het cafe de echte vriendschappen te onderscheiden van de onechte, en stel enkele cruciale vragen aan je beste vriend of Gun Control Laws Should Essay, vriendin. Kijk, meet, leer. Neem je onderwerp, je vraag of the hunchback of notre dame, je twijfel en verzin een onderzoek dat je zou kunnen doen: een literatuuronderzoek in Gun Control Laws be Tightened Essay de bibliotheek, observeer het gedrag van mensen, of room 101 speech, houdt een weeklang je eigen leven in de gaten. Gun Control Should Essay? Kijk, meet, leer. Racism To Stay? Probeer vervolgens hier conclusies uit te trekken, maar liever nog vervolgvragen, argumenten, verdere associaties. Bij het uitwerken van een essay ben je zo vrij als een vogel: het essay heeft geen vaste vorm en geen vaste formule. Gun Control Laws Be Tightened Essay? Je kunt er voor kiezen je gedachtenroute in z’n geheel op te schrijven, en je lezers deelgenoot te maken van je mentale reis.

Maar je kunt ook alleen de resultaten van die reis weergeven, ongeveer zoals Borges zijn ornithologische argument in in America Essay tien zinnen wist weer te geven. Belangrijker nog dan de vorm is be Tightened, de stijl waarin je dit alles giet. The Hunchback Of Notre Dame Quasimodo? Een essay is Gun Control Essay, een literaire vorm en probeer daarom zo weinig mogelijk te preken, ook al geloof je zeer sterk in define contrastive je eigen opvattingen. Gun Control Essay? Wees vooral sceptisch – ook over endgame je eigen gedachten, waarden en ideeen. Onderzoek veel, geloof in Gun Control weinig. Probeer verder algemene uitspraken zoveel mogelijk te concretiseren. John Dalton To Atomic? Dat de verkeersveiligheid eind jaren ’60 bar en boos was, is Should be Tightened Essay, een algemene uitspraak, dat er toen meer dan drieduizend verkeersdoden per contrastive, jaar waren (nu: om en nabij de 1100) is Gun Control Should be Tightened, een concreet feit waar je lezers zich iets bij kunnen voorstellen. Een ander iets dat je bij het schrijven in endgame sparknotes de gaten moet houden is Gun Control Laws Should Essay, dat je natuurlijk met een rationeel betoog bezig bent, met argumenten en redeneringen.

Dat kan een gortdroog betoog worden. Presenteer daarom de feiten zo levendig mogelijk. In America? Wees evocatief: geef je lezer iets om voor z’n ogen tevoorschijn te toveren. Mondialisering is een groot woord, maar een korte beschrijving van de Amerikaanse callcenters in Gun Control Laws be Tightened Essay New Delhi of de bloemenveiling van Aalsmeer waar rozen uit Rwanda worden verhandeld, levert je lezer een beeld. Om het spannend te houden kun je werken met de bekende middelen die je ook bij proza of dalton to atomic theory, toneel toepast: werp vragen op, laat opzettelijk gaten vallen die je elders pas invult en kom steeds terug op je hoofdvraag, het liefst in Gun Control Laws verschillende varianten. Racism Is Here In America Essay? Werk niet met een vast plan, maar wel met een basisidee, een sfeer, een grondtoon. Wat je altijd kunt doen om een langzaam inzakkend essay te versterken is een tweede lijn aan te brengen. Dat kan zelfs een prozaverhaal zijn, zoals Milan Kundera toepaste in Gun Control Should be Tightened Essay zijn novelle Ballingen. Pas met zo’n afwisseling van betoog en verhaal wel op dat je de lezer niet te snel heen en weer laat schieten: geef hem rustig de tijd om in the hunchback of notre je betoog of in je verhaal te komen. Liever een paar pagina’s per Gun Control Laws Essay, deel, dan afwisselen per define, alinea. Leun bij je redeneringen zo weinig mogelijk op de autoriteit van anderen.

Citeren mag, maar nog beter is Gun Control Should, het, om het gelezene te herkauwen en in room 101 speech je eigen woorden neer te pennen. Werk je gedachtenstroom uit. Kies voor een uitgangspunt en bepaal je toon, maar maak geen schema hoe je betoog zal gaan. Laws Should Be Tightened Essay? Laat je leiden door je eigen geschrijf. To Atomic? Zorg er wel voor dat je blijft redeneren, argumenteren, naar voorbeelden en (zintuigelijke) ervaringen zoekt. Ga niet te snel, wees precies in Gun Control Should be Tightened je redeneringen en argumenten. Of Notre Dame? Hoe scherper je gedachtengang kunt beschrijven, hoe dichter je bij nieuwe inzichten komt, zo blijkt uit de ervaring. Tijdens het schrijven ontstaan vaak de interessantste gedachten – geef ze de ruimte. Be Tightened Essay? Lees vervolgens je eigen essay en probeer te achterhalen of john dalton to atomic, hij ‘werkt’. Be Tightened Essay? Waar ga je te snel, waar te langzaam?

Blijft er genoeg over room 101 speech om als lezer door te willen lezen? Is het slot intrigerend genoeg om nog ‘ns flink over na te denken? Laat het geheel vervolgens aan anderen lezen en vraag hen om commentaar. Herschrijf, herformuleer tot je de scherpste route door je eigen geest hebt blootgelegd. Heb je een tip voor een onderwerp van een schrijfoefening? Stuur dan een mailtje hier. Like Schrijven Magazine op Facebook! Door ervaren, professionele redacteuren.

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