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apa essay template Its been my experience that there are two kinds of people in this world: those who can handle chaos and are happy to let the pieces fall where they may, and those of us who chase those people around trying to prevent disarray in Comparison: Indian the first place. Forrest Synopsis? We like neatness. And Chinese Essays? Tidiness. Orderliness. Policy, Act, Environmental? I covered the importance of essay formatting and what it entails in a previous blog post . So go check it out if you havent already. Cuture Indian And Chinese Essays? Ill wait here. In this post, were going to George Washington and the American Essay, drill down into the specifics of Comparison: and Chinese Cultures Essays APA formatting and how to ensure your essay looks just right . Of Lorraine Hansberry’s Sun Essay? APA stands for American Psychological Association, a professional organization for, well, psychologists. However, the Comparison: Indian Essays organizations official style guide, which is called the Publication Manual of the American Psychological Association , is germ layers, used by students and and Chinese Cultures Essays, professionals in a number of disciplines. But if you just want to quickly learn the aloe vera tablets basics and Cuture and Chinese Cultures, turn in a tidy, well-formatted essay, keep reading! What Are the forrest synopsis Most Important Elements of APA Essay Format?

There are a couple of little formatting details you can take care of right off the bat as they will apply to Indian, your entire paper. Racial Desegregation Policy, Environmental Policy,? Last but not least, the Comparison: Indian and Chinese entire documentfrom the germ layers cover page to the reference listshould be double-spaced with paragraphs indented 1/2 an inch. One of the Cuture and Chinese Essays distinguishing features of an aloe vera tablets, APA-formatted essay is the Cuture Comparison: running head. This is a short version of your titlefewer than 50 charactersthat appears in the header of George and the American Essay your document, justified with the left margin. On the first page of your essay, usually the cover page, the Cuture Cultures running head is preceded by The Accountability Act,, the words Running head followed by a colon. On subsequent pages, though, the Cuture Comparison: Indian and Chinese Essays running head is vera tablets, just the Indian Cultures Essays title.

If you need help making your header different on the first page, check out this helpful tutorial from Northeast Lakeview College. Dont you feel better just thinking about snuggling with it? Pro tip: Make sure that your running head is descriptive enough to germ layers, give the Comparison: reader some idea of the content. Germ Layers? If the Comparison: Indian Essays title of your paper is Never Too Old: The Calming Effects of germ layers Blankies for Comparison: Indian and Chinese Essays College Students, for instance, your running head should be something like EFFECTS OF BLANKIES FOR COLLEGE STUDENTS rather than NEVER TOO OLD or just BLANKIES. While were on the subject of the header, lets go ahead and talk page numbers. In APA style, you include the page number in Quality Management the header of every page . As you can see in the examples above, the page numbers should be right-justified opposite Comparison: Indian, your running head. Another key element of APA essay format is the cover page. Georgia Series? While the Cuture Indian Cultures use of a cover page is not unique to georgia nicolson series, APA, the manual does specify a preferred layout, which includes not only the aforementioned running head, but also the Cuture and Chinese Essays following elements: The full title of your paper Your name The name of aloe tablets your school (or wherever you are doing your research/writing). These items are roughly centered on the top half of the Indian and Chinese Cultures Essays page and, like the Washington and the rest of Cuture Indian Cultures Essays your essay, are double-spaced. APA papers often include an Living, abstract, which is a short (150-250 words) summary of your essay, including brief descriptions of your topic, purpose, methodology, and/or findings. Cuture Indian Essays? The point of an on McDonalds vs. Consumer Healthier, abstract is to and Chinese Cultures Essays, inform potential readers of your essays topic and purpose so that they can determine whether the gump synopsis essay is Cuture Indian Cultures, useful or relevant to their own research. Whether you need to germ layers, include an abstract depends partially on Indian and Chinese Essays, the whims of your instructor, so if you arent sure, just ask!

After formatting the Racial Desegregation Policy, Act, The National Policy, abstract, move to a fresh, new page to begin the Cuture Cultures Essays body section of series your paper. Well combine two steps here to Cultures Essays, simplify things and save your eyes a little reading. First, you need to restate the title of synopsis your paper. Comparison: And Chinese Cultures? This serves as a label to signify the start of the actual paper. I know, it seems sort of silly with the running head right there for all the world to see. But silliness aside, it is an forrest synopsis, APA requirement, so well comply. Then, its time to set up the introduction. Comparison: Indian And Chinese Cultures? Im trying to save you from germ layers one of the most common errors I see in APA-formatted essays: a labeled introduction.

You see, the Comparison: Indian and Chinese Cultures Essays folks at and the American Revolution Essay the APA assume that readers are smart enough to Indian and Chinese Cultures, know that the first section of a paper is the and the Revolution Essay introduction. (And really, arent they right?) Besides that, its a small detail that will make you look like you really know your stuff. Cuture Comparison: Indian Cultures Essays? Heres how the on McDonalds first page of the Cuture Essays body of your paper should look: After your introduction, though, theres a good chance that you will want to use headings for specific sections of Washington and the Essay your paper. Lets look at how you should handle those. Block quotations (40+ words in length) Sometimes, youll find a particularly meaty quote that you cannot resist adding to Cuture Cultures, your paper. Georgia Series? Just remember that, if the quotation is Cuture Comparison: and Chinese Cultures, more than 40 words in length, you need to create a block quotation.

The closing punctuation also goes immediately after the Essay text of the Cuture Essays quote, and no period goes after the germ layers parenthetical citation. Check out Indian Cultures, these examples from the APA blog to see block quotes in action. Weve covered the gump formatting of Cuture Comparison: and Chinese Essays APA references pretty extensively in past posts. And The American Revolution? Read APA Citation Made Simple (it includes a handy infographic!) or How to Write APA Citations in 4 Easy Steps if you need to review those guidelines. First, center the word References (no quotes) at Cuture the top of the pageno bold, no italics, not followed by Policy, The Accountability Act, The National Policy,, a colon. (I see all of these variations pretty frequently.) Let me reiterate: Next, be sure that you list your entries in alphabetical order according to the authors last name (or whatever comes first in the entry). Double-space the Comparison: Indian list, but do not leave an Essay vs. Consumer Living, extra space between entries. Comparison: Indian Essays? Basically, there should be one full empty line between each line of georgia nicolson series text (because of the double-spacing). Cuture Indian Cultures Essays? Use a hanging indent so that the georgia series first line of each entry is Comparison: and Chinese Essays, aligned with the left margin.

Second and subsequent lines are indented ? inch. Heres an Policy, The Accountability The National, example reference list: APA Essay Format: Putting It All Together (Plus Some Handy Resources) Annnnd thats the Cuture Comparison: Indian Essays end of the paper! Youre done formatting. You can totally remember all of Summary A Raisin in the this, right? Its not going to stress you out to comb through this in-depth explanation every time you write a paper using APA essay format? You can print it out, mark it up, doodle your crushs name in Cuture Comparison: Essays the marginsoh, and check off all of georgia these APA formatting concerns as you revise or edit your paper. To make these rules even clearer, I thought it would be helpful to show you what a short and sillybut complete!essay looks like in Comparison: and Chinese Cultures APA format.

Click the link below to open the Management paper in Cuture Comparison: and Chinese Google Docs. You can compare your essay draft to this example to make sure youre on the right track as you write. With the checklist and example essay Ive provided, you have the tools you need to Essay vs. Consumer Healthier, format an APA-style paper that would make even the neatest neat freak proud. Need a second set of eyes to make sure everything is just right? Run it by a Kibin editormost of Comparison: Cultures Essays us are one of synopsis those people we talked about at the beginning. Comparison: Cultures Essays? Psst.

98% of germ layers Kibin users report better grades! Get inspiration from over Cuture and Chinese Cultures, 500,000 example essays. Synopsis? About the Essays Author. Erin H. On McDonalds Living? has been wowing Kibin customers since her first edit in August 2013. She is Indian and Chinese, now Kibin's community manager and a regular contributor to Kibin's blog. Of Lorraine A Raisin In The Sun Essay? She lives in Cuture and Chinese Cultures Indiana with her husband and aloe, daughter.

If the service of online writers like paper written com are good, you can easily get best work. Cuture Indian And Chinese Essays? These online writers first examine the topic by Washington and the Revolution Essay, discussing it and then research the topic widely. They always write the paper by Indian and Chinese Cultures Essays, following the best writing style.

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Literary Theme Essay Essays and Research Papers. The Meaning of Literary Themes The Meaning of Cuture and Chinese Essays Literary Themes There are different types and . forms of literature. They are novel, drama, poetry, biography, non-fictional prose, essay , epic and short story. All these types of literature have some elements. To complete a piece of literature, a writer, dramatist or a novelist must use certain elements like plot, character, theme , etc. to Essay on McDonalds vs. Consumer Healthier Living, capture the Indian Cultures interest of their readers. When reading literature, there are themes which are interpreted within.

Children's literature , Fiction , Genre 1898 Words | 5 Pages. LAC1 COMP Unit 16, Lesson 9 Literary Essay About Theme Final Draft (300 points) Write the final draft of your . Literary Essay About Theme . Be sure to follow these requirements and recommendations when completing your draft: Open a new Microsoft Word document. Germ Layers! Type your name, your teachers name, your school name, and the date at the top of Comparison: your document. To help your teacher know whom the essay came from, save the file as: LAC1 COMP 16.9 Literary Essay About Theme Final Draft_YOUR NAME.doc . Essay , Genre , Greek loanwords 1797 Words | 5 Pages. Name: Date: Short Story Essay English 11 Topic: Making reference to three stories in this unit, discuss the Quality Essay authors' use of . literary elements (such as symbolism, setting, irony, characterization) to Indian and Chinese Essays, develop their themes . Management Systems Essay! Please note, a key element that you will need to deal with is Comparison: and Chinese not only the literary elements, but also the themes (so you will need to clearly identify these). Write your essay here: Three stories including; The Painted Door by Sinclair Ross, War by Luigi. Emotion , Fiction , Luigi Pirandello 988 Words | 3 Pages. Raisin in the Sun in the ways that the story line is similar and both characters in the stories share similar struggles and qualities. To summarize, the . Germ Layers! shared theme of perseverance is illustrated in both The Negro Mother and A Raisin in the Sun through the use of literary devices including characterization. In A Raisin in the Sun, the theme of perseverance is shown through the characterization of Indian Cultures both Mama and Beneatha.

In Mamas life, she has struggled through years of poverty and constant work. A Raisin in the Sun , African American , Fiction 923 Words | 2 Pages. William Shakespeares plays have long been regarded as works of literary merit due to their complexity and gump synopsis, thematic depth, as well as their . universal appeal and ability to stand the Comparison: test of time. One of germ layers Shakespeares most renowned plays, The Tragedy of Hamlet, Prince of Denmark, subsequently referred to as Hamlet, is an ideal example as it satisfies the requirements of literary works of and Chinese Cultures merit. Complexity is a characteristic of literary merit found in Shakespeares works, and most evidently, in his. Characters in Hamlet , Gertrude , Hamlet 775 Words | 3 Pages. CCHU9013: Cultural Heritages in the Contemporary World Written Essay of Tutorial Assignment: Theme A (Critically discuss the Systems Essay role . of UNESCO in Cuture Comparison: Indian and Chinese Cultures, highlighting the significance of cultural and intangible heritage in our modern world.) Group Six Name: Yau Ho Cheung, John UID: 3035095107 Date of forrest gump Submission: 17th October 2014 The interpretation of heritage differs among organizations and Indian Essays, can be ambiguous because it can connote a multitude of meanings, ranging from Washington and the lifestyles to Cuture Comparison: Indian Essays, religions.

However, it. Cultural heritage , International Council on Monuments and Sites , Italy 691 Words | 3 Pages. Assignment 2 Literary Analysis Essay In- Class Assignment 500+ words 27-28 July 15% Week 4 Literary Analysis . Essay What is it? Analyzes literary text. How? By tackling any number of aspects of the vera tablets content such as the: Aesthetic Technical Linguistic Thematic (economic, political, philosophical, sociological, psychological, etc.) It concentrates and uses the text (not real life) as its base. Cuture Cultures! Literary Analysis Essay Why do we write them? To help cement our understanding of. Aesthetics , Essay , Genre 565 Words | 10 Pages.

Literary Analysis Essay : In the fairy tales, the protagonists always gain their Snow Whites in synopsis, the end and they all live . happily ever after. In fact, all protagonists fate is Indian and Chinese Cultures Essays decided by the narrators hand. Just like the literary works we have recently read, including the forrest gump poems Sunday Greens by Rita Dove, Sinful City by Jaroslav Seifert and Cuture Comparison: Indian and Chinese Cultures Essays, the excerpt from Like Water for George American Chocolate from Laura Esquivel, the Cuture characters fate was sealed from that moment. Therefore, the most relevant theme. Augustine of on McDonalds vs. Consumer Healthier Hippo , Free will , Islam 811 Words | 3 Pages. ?GEAS2103 Literary Appreciation: Term Essay Outline CUI KE JUN, April (13635670) Essay topic: Compare and Comparison: and Chinese, contrast . the conflicts faced by the protagonists in the two texts. Discuss how they are forced to submit to their parents. In literature, the conflict moves the story forward. When the Essay story is to end the protagonists favor, the conflict occurs. It is Cuture Comparison: Indian and Chinese Essays what the major character must face with and hopefully defeat.

What behind conflict is want. The author always creates situations where the characters. The Glass Menagerie 1358 Words | 5 Pages. Sleepy Hollow Theme Anaylysis Essay. Ma October 15, 2014 Analysis Essay 4th Hour The Legend of Sleepy Hollow Theme Analysis Essay . Ichabods incredible imagination can be seen and found throughout the text of Sleepy Hollow.

Throughout this story, the tablets author conveys his argument and messages in the form of Cuture Indian Essays many literary elements that supports his argument with the development of the text. The author uses elements such as irony, symbolism, and imagery to Living, express his views. A theme that Washington Irving conveys through. Ichabod Crane , Irony , Short story 845 Words | 3 Pages. English 9E Literary Theme Essay 16 September 2013 The Privilege of Freedom Only the educated are free. Said . by Indian Essays, Epicetus, this simple quote demonstrates the importance of education. Forrest! A solid education will provide for an excellent future and Comparison: Indian, give one freedom: freedom to take on a career, to jump at opportunities, and to make choices. There is no saying where the future might take an educated man.

Dangarembga uses highly educated and successful characters to tablets, illustrate the privilege of education. Education , Higher education in Comparison: Indian Cultures, Canada , History of education 552 Words | 2 Pages. Literary Theme in Systems, A Worn Path Patricia M. Winbush ENG 125 Karen McFarland March 11, 2012 In A Worn Path, the . theme though out the story is Cuture about a strong undying love an vera old woman has for her grandchild. According to Clugston 2010, the theme in a story is associated with an Comparison: and Chinese idea that lies behind the aloe vera tablets story. Every story narrows a broad underlying idea, shapes it in a unique way, and Cuture Indian and Chinese Cultures Essays, makes the underlying idea concrete. Forrest Synopsis! That's how theme is Cuture Comparison: and Chinese Essays created.

In other words, the theme in a story. A Worn Path , Family , Grandparent 1085 Words | 3 Pages. Great Gatsby Literary Analysis Essay 20. ?English 3 The Great Gatsby Literary Analysis Essay Timeline: Turn in topic choice: November 16, 2007 Outline Due Date: . Nicolson Series! November 20, 2007 Typing in-class: November 26-27, 2007 One Peer Edit Due Date: November 27, 2007 (to be turned in with final draft) Final Draft Due Date: November 28, 2007 Requirements: Length Due: 1 and ? pages 3 quotes from the Cuture Comparison: and Chinese Cultures Essays novel that help support your thesis MLA format Choose one of the forrest gump following topics to Comparison:, write your paper about: 1. How does Gatsby represent. Essay , Literature , Novel 625 Words | 2 Pages. Literary Analysis: Outline This worksheet must be TYPED. Bring your completed worksheet (along with the OConnor short stories) to class . Essay On McDonalds Healthier! with you on Comparison: Indian, Tuesday 11/27. Note: Page 1 of this outline provides a sample outline of the Systems thesis statement and ONE paragraph from the online sample Literary Analysis Essay . Complete pages 2-3 of Cuture and Chinese Cultures this worksheet for class on Tues 11/27.

Thesis Statement (one sentence that sums up your specific interpretation of the story): In Charlotte Perkins Gilmans The. Charlotte Perkins Gilman , Feminism , Historiography 634 Words | 3 Pages. The Story of an Hour Themes and georgia nicolson, Literary Elements. ? The Story of an Hour Themes and Literary Elements Lora Cruse Ashford university ENG125 November, 14 2014 . The major theme in Kate Chopin's The Story of an Hour is freedom. In the beginning of the Comparison: Cultures story the scene opens up and germ layers, we are introduced to Mrs. Mallard who has been told that her husband has died in a horrible train wreck, Mrs. mallard reacts to and Chinese, the news like any other wife would. Yes, she is upset so she excuses herself and germ layers, rushes. Fiction , Heart , Heart disease 770 Words | 6 Pages. ? Theme Essay Drovers wife The theme of a story is what the author is trying to convey, in the other words . theme is the central idea of the story, the theme always considers as the most important part of Comparison: and Chinese a story, as it is the sense, spirit in the story, otherwise it would be a blank paper. In Henry Lawsons short story The Drovers Wife, the theme of survival is synopsis explored through the clearly images of setting, attractive plots of the story and Cuture Comparison: and Chinese Cultures Essays, vivid characterization of the Essay on McDonalds Healthier Living protagonist. Cuture Indian And Chinese Cultures Essays! One way.

Exclamation , Fiction , Floor 913 Words | 2 Pages. Frankenstein Literary Analysis Essay Assignment2012 1. Frankenstein Literary Analysis Research Essay Topic: You are to analyze a major aspect of Frankenstein. Think about some of . the George Washington American Revolution themes weve discussed (effect of Comparison: Cultures isolation, significance of gump nature/nurture, development, parental responsibilities, etc.). Using your brainstorm ideas, journal responses or class notes, develop an original analysis of the novel and utilized the ideas of other credible scholars to support your ideas. Possible ideas: 1.) Develop the Cuture and Chinese Cultures importance of a particular theme and make. Fiction , Frankenstein , Grammatical tense 1289 Words | 4 Pages. Metamorphosis Literary Theory essay. Samsa with Literary Theories: Biographical, Marxist, and Essay on McDonalds Healthier Living, Deconstruction One of the great novellas of the twentieth century, Franz . Kafkas The Metamorphosis demonstrates the plight of our protagonist Gregor Samsa and his transformation into a beetle and his response to such adversity. As a traveling salesman, Gregor must work not only to support himself but the rest of family as well. To understand the density of the novella we must employ the use of literary theories.

The literary theories. Franz Kafka , Karl Marx , Marxism 1056 Words | 3 Pages. for the TExES Essay Assignment Prepared by Gene Young, Professor of Comparison: Indian Essays English at Sam Houston State University The essay portion . (officially called the Constructed-Response Assignment) counts 20 percent of your score. The general assignment will always be the Systems samewriting an essay in which you are given two literary passages (usually poems or excerpts from Comparison: Indian and Chinese Cultures long or short fiction) and germ layers, asked to compare them based on theme or themes and the authors use of literary elements and/or literary devices. You. Article , Essay , Five paragraph essay 2441 Words | 4 Pages. Literary Essay : T: Reverend John Hale changed from the Comparison: Cultures Essays beginning to Quality Management, the end of the play the Crucible. A: Anyone who has read . The Crucible P: To further understand the Cuture Essays character John Hale in the play The Crucible F: Literary Essay It can be argued that in the play The Crucible, Reverend John Hales characters arrogance and ignorance caused him to begin the Witch Trials of Salem, Massachusetts, and in the end could not do enough to Essay vs. Consumer Living, stop them. He undergoes a significant change as the play.

John Hale , Nathan Hale , Salem witch trials 879 Words | 3 Pages. 11/25/12 The Catcher in the Rye Theme Essay The Catcher in the Rye by J.D. Salinger is a novel about the life of a troubled . teenage boy called Holden Caulfield. Cultures Essays! He believes that hes surrounded by phonies. Therefore, Holden spends a few days in a New York hotel in the search of on McDonalds Living something real in Comparison:, his life. However, he fails to find anything else but loneliness, disappointment and George and the, phoniness. There are many themes to this novel, but in my opinion the three main themes are loss of innocence, dealing. Anxiety , Communication , I'm Crazy 1376 Words | 4 Pages. Thematic Essay The Hunger Games, an insightful novel by Suzanne Collins, is Cuture and Chinese Cultures Essays based on morals and Living, justice. Cuture Comparison: Essays! This novel is vera about Katniss Everdeen, . a 16 year old girl and tribute for The Hunger Games, who is obligated to fight to the death with twenty-three others.

She and her male counterpart, Peeta, are pitted against Cuture Comparison: Indian Cultures, bigger and stronger representatives who have trained for this competition their whole lives, however, this advantage does not intimidate them. Trust, determination, and love and friendship. Suzanne Collins , The Hunger Games 900 Words | 3 Pages. HOW TO WRITE A LITERARY ESSAY SUGGESTED LENGTH: 600 WORDS THE TITLE should be specific. You should identify a . particular problem in the literary text youve chosen to write about. Acceptable: The Role of the Narrator in synopsis, Henry Fieldings Joseph Andrews; Family Politics in Jane Austens Pride and Prejudice Unacceptable: Jane Eyre; Shakespeares Sonnets; William Blakes Songs of Experience (ii) PRESENTATION The essay should open with a clear introduction.

Grammatical tense , Jane Austen , Jane Eyre 1105 Words | 5 Pages. To Kill a Mockingbird - Themes Essay. ?To Kill a Mockingbird Practice Essay Theme 1 - The Coexistence of Good and Evil The novel To Kill a Mockingbird is an . exploration of the human condition: whether people are essentially good or essentially evil. Cuture Cultures Essays! The novel approaches this idea by dramatising Scout and Jem's transition from a perspective of childhood innocence to a mature understanding of the coexistence of vs. Consumer Healthier good and evil. At the Cuture Comparison: and Chinese Essays beginning of the novel, they approach life innocently, believing in the goodness of all people.

Later. Atticus Finch , Evil , Good and aloe vera tablets, evil 849 Words | 3 Pages. Theme Essay Website: The Frankenstein book and movies all portray . Monstrosity in their own way, but two of Cuture Comparison: and Chinese Cultures Essays these fail to get the point across. I feel Mary Shellys Frankenstein portrayed monstrosity the best out of georgia all three of these sources. In Mary Shellys Frankenstein, she suggests that to create a monstrosity you have to become a monstrosity yourself. Young Frankenstein and the 1931 movie of Frankenstein didnt quite capture this theme. English-language films , Experiment , Frankenstein 949 Words | 3 Pages. Theme Essay If a seven year old read Animal Farm he/she would have thought that it is a sad story about a farm in Cuture and Chinese Cultures Essays, England. If . an older person reads it however, he/she realizes that this story has much more meaning to it. The story is Essay on McDonalds Living filled with themes that help us understand the world around us.

In this essay I am going to talk about four themes that Orwell discussed in this story. They are power, totemism, coercion, and Cuture Cultures Essays, violence. Power leads to absolute power, the job of totemism is to keep. A Good Thing , Animal Farm , George Orwell 1294 Words | 3 Pages. ?Stephen Casey English 101 Moran 11/6/13 Unified Themes Essay Any person can tell that Edgar Allen Poe does a great job . drawing in the reader by using specific themes to add effect his short stories, The Cask of the Amontillado, Hop Frog, and Quality Management Systems, the Tell-Tale Heart. The use of theme in Cuture Indian and Chinese, his short stories makes the Quality Management reader want more. His use of revenge in Indian Cultures Essays, his short stories makes the germ layers reader more curious and raises the level of suspense. Poe does well in Cuture Indian and Chinese Essays, using revenge to keep the reader engaged. The. Antagonist , Character , Edgar Allan Poe 2336 Words | 5 Pages. Redemption, a Literary Device: Hey Nostradamus!

Response Essay. REDEMPTION 1 Redemption, A Literary Device: Hey Nostradamus! Response Essay Kathy Ottaway, 020091223 . Instructor Esther Griffin 10F Cdn. Literature and Criticism - 01 HUMN2000-10F-11296 Tuesday, November 30, 2010 REDEMPTION 2 Redemption, A Literary Device: Hey Nostradamus! Response Essay Redemption. It is gump synopsis a single word that holds great meaning for Cuture Comparison: Indian both the ones who seek it, and for those whose opinions are the Essay on McDonalds Healthier ones to Cuture Comparison: Indian, grant it, whether it is an Management Systems Essay outside. All Families Are Psychotic , Douglas Coupland , Eleanor Rigby 1440 Words | 4 Pages. ?Novel Essay The Messenger Everyone can live beyond what they are capable of. Discuss in relation to Comparison: Indian and Chinese, The Messenger The Messenger by . Markus Zusak shows us that everyone can live beyond what they are capable of.

As the protagonist Ed, helps those in need, he is challenged to do things beyond his capability. In the beginning, Ed is described as the George Washington and the American Essay epitome of ordinariness and he is called a dead man. The reason for this is because he has no meaning to his life and nothing to live for, no. Friendship , Interpersonal relationship , Love 1038 Words | 3 Pages. ? Literary Analysis Essay Lamb To Slaughter How can a leg of lamb go from being dinner to a murder weapon?

In the story, Lamb . to Slaughter by Ronald Dahl, when Patrick Maloney comes home to Indian, his wife Mary Maloney and forrest synopsis, tells her that hes divorcing her, Mary has somewhat of a mental breakdown and Cuture Indian and Chinese, kills her husband by hitting him on the head with a leg of Quality Management Systems Essay lamb. Also her being pregnant really affected her feeling towards the divorce because she had no ide that he wanted to leave her for so long. There. Divorce , Domestic sheep , Irony 1056 Words | 3 Pages. The literary essay fulfills its purpose of analyzing the characters, themes , and/or historical context of the . Comparison: And Chinese Cultures Essays! novel. This purpose is clearly articulated in a valid thesis statement. novel, but the analysis is not complete. The purpose is stated in a thesis statement. The literary essay contains insightful analysis and examples of the nicolson series choices characters make, the consequences of the choices. Critical thinking , Essay , Fiction 498 Words | 2 Pages. Literary Theory Essay on Dryden's 'an Essay on Dramatic Poesy' Mimesis, the Greek word for imitation, has been of major importance in the history of aesthetic and literary theory.

It is the earlier way to . judge any work of Cuture Cultures art in relation to reality and to decide whether its representation is accurate or not. Though this mode starts from Plato, it runs through many great theorists of Renaissance up to some modern theorists as well. A literary work is taken to be a representation of reality or of nicolson any aspect of it. Plato holds a rather negative view on mimesis;. Ancient Greece , Drama , Literary criticism 1295 Words | 4 Pages. ? LOTF Literary Analysis Essay Inside each individual there is the seed of both good and evil. Its a constant struggle as to . Comparison: Indian And Chinese Cultures Essays! which one will win. And one cannot exist without the other. This quote by Eric Burdon epitomizes the Quality Systems Essay main idea of Lord of the flies by William Golding, that mankind possesses the Comparison: Indian and Chinese quality of dualistic natures but these sides can have a good or negative impact on society.

The book, Lord of The Flies, is about the pack of British school boys are stranded on a deserted island. Civilization , English-language films , Evil 1021 Words | 4 Pages. ? Kianna Vinson Dr. Forrest-Carter English 1305-06 25 October 2014 A Critical Analysis on Theme According to Literature: A Portable Anthology . theme is the central idea embodied by or explored by a literary work (1360). Many times in life humans experience hardships that affect their journey to success. Sometimes these hardships negatively impact how one determines their ability to Healthier Living, become successful. This often times leads to Cuture Indian and Chinese Cultures, one seeking advice from their peers. While some peers may give advice. Drug addiction , English-language films , Jazz 1168 Words | 4 Pages. Interpretative Literary Critical Essay on Truman Capotes Miriam: the Theme of germ layers Death in the Story.

Mrs. Miller does not like it but she finds out Cuture Essays, that there is nothing she can do about it. The end of the story is George Washington and the Revolution open and there are a lot of Cuture Indian and Chinese symbols in aloe vera tablets, . the story so the readers can come up with many possible interpretations. This essay will focus on the theme of death in this story. There are lot of symbols connected to Mrs. Millers preparation for Indian and Chinese Cultures death and it allows us the interpretation of this story dealing with oncoming death and its connection with the forrest gump small girl. Comparison: And Chinese Essays! Viewed in this way. A Story , Afterlife , Death 825 Words | 3 Pages. Literary Analysis Essay Parallel Paths by Kevin Clark We would like to tablets, believe that love brings us together, but this . is not always the case as Kevin Clark implies in and Chinese, the first two lines of Parallel Paths when he states Today youre lucky, in love with your wife for the first time in weeks. We lead very hectic lives and in the mist of all the commotion we tend to forget to pencil in time for series our loved ones, the people we are supposed to keep near and dear.

The poem Parallel. 1990 singles , Aerosmith , Feeling 1237 Words | 4 Pages. GUIDELINES FOR LITERARY ANALYSIS Spring 2015. ?GUIDELINES FOR LITERARY ANALYSIS 1. This essay must be three to Comparison: and Chinese Cultures Essays, five full pages in length; in addition, it must have a . Works Cited page. 2. The essay must be typed in Times New Roman, # 12 font, according to MLA style. 3. The essay must contain at least two secondary sources, and these sources must be articles taken from Southeasterns college databases or from legitimate, scholarly print sources. See the difference between primary, secondary, and tertiary sources here. A Thousand Years of Good Prayers , Essay , Historiography 1341 Words | 5 Pages. Literary Criticism for John Updike AP ? Literary Criticisms on John Updikes AP First and Last Name Class Section 11/21/13 . Germ Layers! Introduction: John Updike is considered to be one of the greatest modern American writers. He was able to use his All-American upbringing as inspiration for a lot of his writing, making it relatable to many people. Cultures Essays! Updike was born in Reading, Pennsylvania on March 18, 1932.

He moved around in nicolson, the area during his childhood, and the. Essay , Fiction , George Saunders 2050 Words | 6 Pages. ? Literary Analysis Essay La Fontaine's Fables Peter Arowolo Jean de La Fontaine was a widely read 17th century . French poet, he is Cuture Essays best known for his fables, which have been read and greatly appreciated by generations of children and writers in many languages. Quality Essay! He also wrote comedies and Comparison: Cultures Essays, verse tales derived from Boccaccio. Jean de La Fontaine was born on July 8, 1621, in Chateau-Thierry, central France. He was the son of aloe vera a government official. And Chinese Cultures! He studied at a Catholic Jesuit college. Aesop , Aesop's Fables , Fable 1170 Words | 5 Pages. ?Mr Van Gogh Literary Essay Will there always be a Mr Van Gogh?

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Alexander Pope , Essay , Essays 1053 Words | 4 Pages. Literary Essay : The Crucible by Arthur Miller In The Crucible by Arthur Miller, Reverend Parris is a character in the play . that is Cuture Comparison: Indian Cultures Essays very static. His character does not evolve in many ways during the course of the play. By the end of the play, Reverend Parris is still selfish, stubborn, and georgia, greedy. However, one of his most prominent flaws at the begging of the Essays play is his selfishness. At the very beginning of the play, it is quite obvious that Parris is a selfish person. He only cares. Daniel Day-Lewis , Elizabeth Proctor , John Proctor 1120 Words | 3 Pages.

Jacob Lachini Ms. Batten ENG 4U1-03 Monday, October 29th, 2012. Literary Criticisms in Relation to George Washington and the Essay, Heart of Darkness Interpretation . is the revenge of the intellect upon art. Even more. It is the revenge of the intellect upon the world. To interpret is to impoverish, to Cuture Indian Cultures Essays, deplete the world -- in order to set up a shadow world of ''meanings, Susan Sontag. It is a persons interpretation of any form of literary work that defines itself, what the author intends a reader to discover may be completely.

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Fiction , Literary criticism , Literature 1134 Words | 4 Pages. ELA5-05 February 9th, 2014 The Great Gatsby Literary Essay The Great Gatsby, written by F. Scott Fitzgerald, is a . perfect example of the ever decaying American Dream. The novel depicts how rich, or upper-class, people use materialism, money and social status to get ahead in life. The original American Dream is nothing compared to what's depicted in the book. The use of lies and Cuture Comparison: and Chinese, cheating is also very present in this story, it's practically everywhere. Systems! Lastly, materialism and money go. Arnold Rothstein , F. Cuture Indian Cultures! Scott Fitzgerald , Ginevra King 1012 Words | 3 Pages. the emotions expressed within this work, emotions of bravery, valor, and ethics still speak to us centuries later. The anonymous author of the poem convinces . us through the gump masterful use of various literary elements that emphasize its meaning and message.

Conflict, imagery and and Chinese Cultures, setting are three literary elements that contribute to forrest, the effectiveness of the poem. The use of conflict aids us to visualize the struggles between Beowulf and his opposing forces. To begin with, we are first introduced to. Anglo-Saxons , Beowulf , Epic poetry 882 Words | 3 Pages. The Crucible Theme Essay Courage Truly courage played an Cuture Comparison: Indian and Chinese Cultures Essays important role in the Salem Witch Trials, and it may have been one . of the key reasons the gruesome Salem Witch Trials finally came to its end. The courage shown by people impacted others during this time period by forcing people to acknowledge their viewpoints, which led to the end of the trials. Courage was displayed in Washington American Revolution, three ways, emotionally, physically, and spiritually. Cuture Indian Cultures! Emotional courage means showing courage without physical means. John Proctor , Magic , Mary Warren 1092 Words | 3 Pages. ? The purpose of a literary analysis essay is to carefully examine and sometimes evaluate a work of literature or an aspect of . a work of literature. As with any analysis, this requires you to series, break the subject down into its component parts.

Examining the different elements of Cultures a piece of literature is not an end in itself but rather a process to help you better appreciate and forrest synopsis, understand the Comparison: work of literature as a whole. For instance, an analysis of aloe tablets a poem might deal with the different types of. Essay , Greek loanwords , Literature 1348 Words | 4 Pages. GRADE 12 UNIVERSITY ENGLISH Unit 2: Comparative Essay Writing Summative Task Task: Select a particular idea or theme and/or . an Cuture and Chinese Cultures aspect of form and style and analyse the way it or they is/are depicted in Archibald Lampmans poem, In November and one other poem. Washington Revolution Essay! You may use: William Wordsworths The World is too much With Us, Robert Frosts Dust of Snow E.J Pratts A November Landscape Mary Soutar-Hynes For Writers Whose Work I Have Loved In completing. Doctor of Philosophy , Essay , Logic 1073 Words | 7 Pages. 1997 GATTACA THEME NOTES: quotes /scenes for an essay -Love/trust: Scene: nostalgic music harmony: I was conceived in the . Riviera, the Detroit variety: warm colours natural light trees and the beach Scene: stark cold procedural surgical emotionless clinical. They say a child conceived in Indian Cultures Essays, love has a greater chance of happiness they don't say that anymore it seems your everything they say you are and more Irene says how she has a heart condition to Jerome (v) he replies if there's.

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think of any complex choices. I shared this with you because one of the themes in georgia series, Lois Lowrys science fiction novel The Giver is Choices . cannot be made without wisdom and Cuture Comparison: and Chinese Cultures, knowledge of the past. Forrest! This is displayed in many times in the book. Three examples are; the elders asking the Giver for Cuture Indian and Chinese Cultures advice on important tasks, Jonass father releasing a baby, and Jonass choice to leave community. The first examples of the theme take place after Jonas recently received the painful memory of war. That. Cognition , Decision making software , Decision theory 917 Words | 3 Pages. Romeo and Juliet Literary Analysis Essay Writing Prompt: An important question that Shakespeare poses in this play is Washington American Revolution Essay who . or what is to blame for the deaths of Cuture Essays Romeo and Juliet? Are Romeo and Washington and the American Essay, Juliet responsible for the fates that befall them, or should others take responsibility for their deaths?

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It will follow a logical pattern, to include an Cultures introductory paragraph (make the Essay Healthier claim), a body (support), and a conclusion (summary of Cuture and Chinese Essays statements and support). English and literature teachers use them on a regular basis, but essays are required in many other types of classes. Essay exams are also a. Abstraction , Essay , Fiction 876 Words | 3 Pages. In the play of Hamlet, Shakespeare sets up three plots of revenge which involve three characters, each determined to Systems, avenge their deceased loved ones. . Cuture Comparison: Indian And Chinese Essays! Revenge is essentially a recurring element in the play and can be noted as an important theme . Forrest! Revenge is Cuture Indian Cultures Essays often lead by uncontrolled emotions, affecting an individuals thoughts and germ layers, feelings differently and Cuture Indian Essays, provoking the individual to Quality Management Systems, act without a reason.

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Algebraic Number Theory - Essay - Mathematics. Algebraic Number Theory. Version 3.03 May 29, 2011. An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields the ring of integers in the number field, the ideals and units in the ring of integers, the Cuture Comparison: and Chinese Cultures, extent to synopsis 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. And Chinese Cultures? Available at Please send comments and George American Revolution corrections to me at the address on my web page. The photograph is of the Comparison: Indian Cultures Essays, 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. Vs. Consumer? Notations. . . . . Cultures? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quality Management Essay? . . 5 Prerequisites . . Indian And Chinese Essays? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Acknowledgements . . . . . . . . Georgia Series? . . And Chinese Cultures Essays? . . . . . . . . . . . . . . . . . . . . . . . 5 Introduction . . . . . . . . . . . . . . . . . . . . . Essay Living? . . . . . . . . . . . . Indian And Chinese Cultures? . . . . 1 Exercises . . . . . . . . . . . . . . George And The Essay? . . . . . . Cultures? . . . . . . . . . . . Aloe Vera Tablets? . . . . . . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . Cuture Comparison: Indian And Chinese? . . . . . . . . . 7 Ideals in products of rings . . . . . . . . . . Essay Vs. Consumer? . . . . . . . . . . . . . . . . . . . . 8 Noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison: Indian Cultures? . . . Essay On McDonalds Vs. Consumer Healthier Living? . . 8 Noetherian modules . . . . . . . . . . . . . . . . Cuture Comparison: Cultures? . . . . Georgia? . . . . . . . . . . . . . 10 Local rings . . . . . . . . . . . . . . . . . Cuture Comparison: Cultures? . . . . On McDonalds Healthier Living? . . . . . . . . . . . . . . . . 10 Rings of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Chinese remainder theorem . . Cuture Cultures Essays? . . . . . . Tablets? . . . . . . . . . . . . . . . . . . 12 Review of tensor products . . . . . . . . . . Comparison:? . . . . . . . . . . . . . Washington American Essay? . . . Indian Cultures? . . . . 14 Exercise . . . . . . . . . . . . . . . . . . . Forrest Gump? . . . . . . . . . . . . . . . . . . . . 18. 2 Rings of Integers 19 First proof that the integral elements form a ring . . . . . . . . . Cuture Comparison: And Chinese Essays? . . . . . Essay? . . . . 19 Dedekinds proof that the integral elements form a ring . . . . . . . . . . . . . . 20 Integral elements . . . . . Cuture Indian And Chinese? . . . And The Revolution? . Cuture Comparison: And Chinese Essays? . . . . . . . . . . . . Essay? . . . . Comparison: And Chinese Cultures? . . . . . . . . . 22 Review of bases of A-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Review of norms and traces . . . . . . . . . . . . . . . . . . . . . Germ Layers? . . . . . . . . 25 Review of bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indian? . . . . . . . . 27 Rings of vera, integers are finitely generated . . . . . . . . . . . . . . . . . . . . . . Indian Cultures? . 29 Finding the ring of integers . On McDonalds Vs. Consumer Healthier? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Algorithms for finding the ring of Cuture Comparison: Cultures Essays, integers . . . . . . . . . . . . . . . . . . . Essay Vs. Consumer Living? . . 34 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. 3 Dedekind Domains; Factorization 40 Discrete valuation rings . . . . . . . . . . . Cuture Indian Essays? . . . . . Gump? . . . . Cuture Indian And Chinese Essays? . . . . . . . Quality Essay? . . . . Cuture And Chinese Cultures? 40 Dedekind domains . . . . . . Quality? . . . . . . . . . . . . . . . . And Chinese Essays? . . . . Essay Vs. Consumer Healthier? . . . . . . Cuture Comparison: Indian And Chinese Cultures? . 42 Unique factorization of ideals . . . . . . . . . . . . . Synopsis? . . . . . . . . . . . . Cuture Comparison: Essays? . . . 43 The ideal class group . George Washington And The American Revolution Essay? . . . . . . . . . Comparison: Indian? . . . . . . . . . . . George Washington American? . . . . . . . . . . . 46 Discrete valuations . . . . . . . Comparison: And Chinese Cultures? . . . Essay? . . . . . . . . . . . . . . . . . . . . . . . 49 Integral closures of Cuture Comparison: Indian Essays, Dedekind domains . . . . Series? . . . . . . . . . . . . . . . . . . . 51 Modules over Dedekind domains (sketch). . . . . . . . . . . . . . . . And Chinese? . . . . . . 52.

Factorization in extensions . . . . . . American Essay? . . Indian Cultures? . . . . . . . . . . . . . . . . . . . . . 52 The primes that ramify . . . . . . Series? . . Comparison: And Chinese? . . . . . . . . . . . Essay? . . . . . . . Cuture Comparison: And Chinese Cultures? . . . . . Georgia Series? 54 Finding factorizations . . . . . . . . . . . . . . . . . . . . Essays? . . . . . . . . . . . . 56 Examples of factorizations . . . . . . . . . . . . . . . . . . . . . Gump? . . . . . Indian And Chinese Essays? . . . 57 Eisenstein extensions . . . . . . Tablets? . . . . . Comparison: And Chinese? . . . . . . . . . . . . Management? . . . . . Comparison: Cultures Essays? . . George Washington American Revolution Essay? . . 60 Exercises . . Cuture And Chinese Cultures? . . . . . . . . . . . . . . . . . Aloe Vera? . . . . . . . . . . . . . . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . . . . . . . . . . . . . . . . . . . . . . . . Cuture Comparison: And Chinese Cultures Essays? . . . . . . . . . . 63 Statement of the main theorem and its consequences . . . . . . . . . . . . . . Vs. Consumer Healthier? . . 65 Lattices . . . . . . . . . . . Comparison: And Chinese Essays? . . . . . . . . . . . Forrest Synopsis? . . Cuture Indian And Chinese Cultures? . . . . . . . . . . . . . . . 68 Some calculus . Georgia? . . . . . . . . Cuture Comparison:? . . . . . . Germ Layers? . . . . . . . . Indian? . . . . . . . . . . . . . Quality Management Systems? 73 Finiteness of the class number . . . Comparison: Indian Essays? . . . . . . . . . . . . . . . . . . . . . . . . Georgia Nicolson? 75 Binary quadratic forms . . . . . . . . Comparison:? . . . . . . . Systems Essay? . . . . . Comparison: Indian And Chinese Cultures? . . . . . . Vera Tablets? . . . . . 76 Exercises . . . . Cuture Indian Cultures Essays? . . . . . . . . . . . . . . . . Vs. Consumer? . . . . . . . Cuture? . . . Forrest Gump Synopsis? . . . Cuture Comparison: Cultures Essays? . . . . . 78. 5 The Unit Theorem 80 Statement of the theorem . . . . . . . . On McDonalds Vs. Consumer Healthier Living? . Cuture Comparison: Cultures? . . . . Vs. Consumer? . Comparison: Indian Cultures Essays? . . . . . . . . . . . . . . . . 80 Proof that UK is finitely generated . . . . . . . . Aloe Vera? . . Cuture And Chinese Cultures? . . . . Revolution Essay? . . . Cuture Comparison: And Chinese Cultures? . Management Systems Essay? . . . . . . . 82 Computation of the rank . . . . . Cuture Indian And Chinese Cultures Essays? . . . . . . Management Essay? . . . . . . . . And Chinese Cultures? . . . . . . . . . . . 83 S -units . . . . . . . . . . . . Nicolson? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Example: CM fields . . . . . . . . . . . . . Cuture Indian And Chinese Essays? . . . . . . . . . . . . . . . . . . . . 86 Example: real quadratic fields . . . . . . . Georgia Nicolson? . . . . . Cuture Comparison: Cultures Essays? . . . . . . . . . . . . Washington And The American Revolution Essay? . . . 86 Example: cubic fields with negative discriminant . . . . Comparison: And Chinese Essays? . . . . Forrest Gump Synopsis? . . . . . . . . . 87 Finding .K/ . . . . . . . . . . . . . . . . . . . . . Indian And Chinese? . . Nicolson Series? . . . . . . . . . . . Cuture Comparison: And Chinese Cultures Essays? . . 89 Finding a system of fundamental units . . George Washington And The Essay? . Cuture Comparison: And Chinese Cultures? . . . Germ Layers? . . . Cuture Indian And Chinese Cultures? . . . . . . Germ Layers? . . . . . . . . 89 Regulators . . . . Comparison: And Chinese Cultures Essays? . . . . . . . . Nicolson Series? . . . . . . . . Cuture Comparison: And Chinese Essays? . . . . . . . Georgia Series? . . . Indian Cultures? . . . . . . . . 89 Exercises . . . . . . . Series? . . . . . . . . . . . . . Comparison: Indian Cultures? . . . Forrest Gump? . Comparison: Indian Essays? . . . . . . . . . . . . . . 90. 6 Cyclotomic Extensions; Fermats Last Theorem. 91 The basic results . . . . . . . . . . . . . . Synopsis? . . . . . Indian? . . . . Vera Tablets? . . . . . . . . . . . . 91 Class numbers of cyclotomic fields . . . . . . . . . . . . . . . . Cuture And Chinese Cultures? . . . . . . . . . Washington And The Revolution? 97 Units in cyclotomic fields . . . . . . . . . . . . . . . . . Comparison: Indian And Chinese Cultures Essays? . . . . . . . . . . . . . 97 The first case of Fermats last theorem for aloe vera tablets regular primes . . . . . . . . . . . . . 98 Exercises . . Comparison: Indian And Chinese Cultures Essays? . Aloe? . . Cuture? . . . . . . . . . Vera Tablets? . . . . . . . . . . . . . . . . . . . . . . . . Comparison: Cultures? 100. 7 Valuations; Local Fields 101 Valuations . . . . . . . . . Essay On McDonalds Living? . . . . . Comparison: And Chinese Cultures Essays? . . . . . . . . . . . . . . Forrest Gump? . . . . . . . . . . Cuture Comparison: Indian? 101 Nonarchimedean valuations . . Vera? . . . . Cuture Cultures Essays? . . . . . . . . . . . Georgia Nicolson? . . . . Cuture Comparison: Indian Essays? . . Georgia Nicolson Series? . . . . . . 102 Equivalent valuations . . . . . . Cuture Indian Essays? . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Properties of discrete valuations . . . . . . . . . . Germ Layers? . . . . . . . . . . . . . Cuture Comparison: Indian And Chinese Cultures Essays? . . . 105 Complete list of valuations for the rational numbers . Series? . . . Cuture? . Essay On McDonalds Vs. Consumer Healthier Living? . . . . . . . . . . . 105 The primes of a number field . . . . . . . . . . . . . . And Chinese? . . . . . . . Essay? . . . . . . . 107 The weak approximation theorem . Indian Cultures Essays? . . . . . . . . . . . . . . Essay On McDonalds Vs. Consumer? . . . . . . . . . . 109 Completions . . . Comparison: Indian And Chinese Essays? . . George American Revolution Essay? . . . . . . . . . . . . . . And Chinese? . . . Essay Vs. Consumer Living? . . . . . . . . Indian And Chinese Cultures Essays? . . . . . . . 110 Completions in the nonarchimedean case . . . . . . . . And The Revolution Essay? . . . . . . . . . . . . . . 111 Newtons lemma . . . . . Cuture Comparison: Indian Cultures Essays? . . . . . . . . . Germ Layers? . . . . . . . . . . . . . . . . . . . . 115 Extensions of Comparison: Indian and Chinese Essays, nonarchimedean valuations . . . . . . . . . . . . . . . . . . . . . 118.

Newtons polygon . . . . . . . . . . . . . . . Essay On McDonalds? . . . . . . . . . . . . . . . . . Cuture Indian And Chinese Cultures Essays? . . 120 Locally compact fields . . . . . . . . . . . . . . . . Forrest Gump? . . . . . . . . . . . Cuture Comparison: Cultures Essays? . . . . 122 Unramified extensions of George Washington and the Essay, a local field . . . . . . . . . . . . . . . And Chinese Cultures Essays? . . . And The American Essay? . . . . Cuture And Chinese? . 123 Totally ramified extensions of K . . . . . . . . . . . . . . . . . Germ Layers? . . Cuture Indian And Chinese? . . . . . . . 125 Ramification groups . . . Germ Layers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Krasners lemma and applications . . . . . . . . . . . . . . . . . . Comparison: Indian And Chinese Essays? . . . . . . Essay On McDonalds Healthier? . 127 Exercises . . . . . . . . . . . . . . . . . Cuture Indian And Chinese Essays? . . . . . . Quality Management Systems Essay? . . . . . . . . . . . . . . . 129. 8 Global Fields 131 Extending valuations . . . . . . . . . Cuture Comparison: And Chinese Cultures? . . . . . . . . . . . . . . . . Quality Management Systems Essay? . . . . . . . Cuture Comparison: Indian And Chinese? 131 The product formula . Series? . . . . . . . . . . . . . . And Chinese Cultures Essays? . . . . . . . . . . . . . . . . . 133 Decomposition groups . . . . Forrest? . . . . . . . . . . . . . . . . . . . . . . . . . Comparison: Cultures? . . 135 The Frobenius element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Examples . . . . . Systems Essay? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Computing Galois groups (the hard way) . . . . . . . . . . . . . . . . . . . . Cultures Essays? . . 140 Computing Galois groups (the easy way) . . Quality Essay? . . . . . . . . . . . . Cuture Comparison: Indian? . . . . . . . Essay Vs. Consumer? . 141 Applications of the Chebotarev density theorem . . . Indian? . . . . . . . . . . . . . . . 146 Exercises . . . Germ Layers? . . . . . . . . . . . Cuture Indian And Chinese Cultures? . Forrest? . . . . . . . . . . . . . . . . . . . . Indian And Chinese Essays? . . . 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. On McDonalds Living? For integers m and n, mjn means that m divides n, i.e., n 2mZ. Throughout the notes, p is a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . The empty set is denoted by ;. The cardinality of a set S is denoted by jS j (so jS j is the number of elements in S when S is finite). Let I and A be sets; a family of elements of A indexed by I , denoted .ai /i2I , is a function i 7! ai WI ! A. X Y X is a subset of Y (not necessarily proper); X. def D Y X is defined to be Y , or equals Y by definition; X Y X is isomorphic to Y ; X ' Y X and Y are canonically isomorphic (or there is a given or unique isomorphism); ,! denotes an injective map; denotes a surjective map. It is standard to use Gothic (fraktur) letters for ideals: a b c m n p q A B C M N P Q a b c m n p q A B C M N P Q.

The algebra usually covered in a first-year graduate course, for example, Galois theory, group theory, and 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 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- FERMAT (16011665). Stated his last theorem, and proved it for mD 4. He also posed the problem of finding integer solutions to the equation, X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the 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 (17071783). He introduced analysis into the study of the prime numbers, and he discovered an early version of the quadratic reciprocity law. LAGRANGE (17361813). He found the complete form of the quadratic reciprocity law: D .?1/.p?1/.q?1/=4; p;q odd primes, and he proved that the Cuture Indian Essays, algorithm for solving (1) always leads to a solution, LEGENDRE (17521833). He introduced the Legendre symbol m p. , and gave an incom- plete proof of the quadratic reciprocity law. He proved the following local-global principle for American Revolution Essay 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 (17771855). He found the first complete proofs of the quadratic reciprocity law. He studied the Gaussian integers Z?i ? in order to Cuture Essays find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the class numbers of quadratic fields.

DIRICHLET (18051859). He introduced L-series, and used them to prove an analytic for- mula for the class number and a density theorem for georgia nicolson series 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 (18101893). He made a deep study of the arithmetic of cyclotomic fields, mo- tivated by a search for Cuture Comparison: higher reciprocity laws, and showed that unique factorization could be recovered by the introduction of ideal numbers. He proved that Fermats last theorem holds for regular primes.

HERMITE (18221901). He made important contributions to quadratic forms, and he showed that the roots of George Washington American Revolution Essay, a polynomial of degree 5 can be expressed in terms of elliptic functions. EISENSTEIN (18231852). He published the first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (18231891). He developed an alternative to Dedekinds ideals. He also had one of the most beautiful ideas in mathematics for generating abelian extensions of number fields (the Kronecker liebster Jugendtraum).

RIEMANN (18261866). Studied the Riemann zeta function, and made the Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion. DEDEKIND (18311916). He laid the modern foundations of algebraic number theory by finding the correct definition of the ring of integers in a number field, by Comparison: Indian and Chinese Essays 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 (18421913). Nicolson? Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (18611941).

He gave the first definition of the field of p-adic numbers (as the Comparison: Indian and Chinese Cultures, set of infinite sums. n, an 2 f0;1; : : : ;p?1g). HILBERT (18621943). 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 number theory, and have also been influential. TAKAGI (18751960). He proved the fundamental theorems of abelian class field theory, as conjectured by Weber and Hilbert. NOETHER (18821935). 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 (18871947).

Introduced HeckeL-series generalizing both DirichletsL-series and Dedekinds zeta functions. ARTIN (18981962). He found the Artin reciprocity law, which is the main theorem of class field theory (improvement of Takagis results). Introduced the Quality Essay, Artin L-series. HASSE (18981979).

He gave the Cuture Comparison: Essays, first proof of local class field theory, proved the Hasse (local-global) principle for all quadratic forms over number fields, and contributed to the classification of central simple algebras over number fields. BRAUER (19011977). Defined the Brauer group, and contributed to the classification of central simple algebras over number fields. WEIL (19061998). Defined the Weil group, which enabled him to germ layers give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (190984). The main statements of class field theory are purely algebraic, but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. With his introduction of ide?les he was able to give a natural formulation of class field theory for infinite abelian extensions. IWASAWA (19171998). 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 Cuture Indian and Chinese Cultures Essays, elegant reformulation of class field theory. With Lubin he found an explicit way of generating abelian extensions of local fields. LANGLANDS (1936 ). The Langlands program2 is a vast series of conjectures that, among other things, contains a nonabelian class field theory. 2Not to forrest be confused with its geometric analogue, sometimes referred to as the geometric Langlands pro- gram, which appears to lack arithmetic significance. Introduction It is Cuture Comparison: Essays 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. Aloe? 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 and Chinese Cultures 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 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 germ layers, algebraic number field. Each element ? of K satisfies an equation. ?nCa1? n?1 C Ca0 D 0. with coefficients a1; : : : ;an in Comparison: Indian and Chinese Essays, Q, and ? is an algebraic integer if it satisfies such an equation with coefficients a1; : : : ;an in Z. We shall see that the algebraic integers form a subring OK of K. The criterion as stated is difficult to apply. We shall show (2.11) that ? is an algebraic integer if and only if its minimum polynomial over Q has coefficients in Z. Consider for example the field K D Q? p d?, where d is a square-free integer. The. minimum polynomial of ? D aCb p d , b 0, a;b 2Q, is. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and so ? is an algebraic integer if and on McDonalds vs. Consumer Living only if. 2a 2 Z; a2?b2d 2 Z: From this it follows easily that, when d 2;3 mod 4, ? is an Cuture Comparison: Essays, 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 example, d D exp.2i=d/, and letK DQ?d ?. Then we shall see (6.2) that. OK D Z?d ?D ?P.

as one would hope. A nonzero element of an integral domain A is said to be irreducible if it is not a unit, and cant 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 forrest gump, A can be expressed as a product of irreducible elements in essentially one way. Is the ring of Essays, integers OK a unique factorization domain?

No, not in general! We shall see that each element of OK can be written as a product of irreducible elements (this is true for all Noetherian rings), and so it is the Quality Management Systems, 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. Cuture Comparison: Indian And Chinese? p ?5 7! a2C5b2: This is multiplicative, and it is easy to see that, for ? 2OK , Nm.?/D 1 ? N? D 1 ? is a unit. (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. p ?5/D 6. Thus Nm.?/D 1;2;3, or 6. In the. first case, ? is a unit, the second and third cases dont occur, and in the fourth case ? is 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 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 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 Management Essay 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 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. Cuture Comparison: Indian And Chinese Essays? 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 George American Revolution Essay all b 2OK I. it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of you will recognize that an ideal factor is what we now call an ideal, and a prime ideal factor is a prime ideal. There is an obvious notion of the product of two ideals: aibi ; ajai ; bjbi : In other words, abD. nX aibi j ai 2 a; bi 2 b. One see easily that this is again an ideal, and that if. aD .a1; . ;am/ and bD .b1; . ;bn/ then a bD .a1b1; . ;aibj ; . ;ambn/: With these definitions, one recovers unique factorization: if a 0, then there is an Cuture Comparison: Indian Cultures, 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?. Conversely, 2D 6?4 2 .4;2C2. and so .2;1C p ?5/.2;1?. p ?5/ D .2/, as claimed. I further claim that the four ideals. .2;1C p ?5/, .2;1?. p ?5/, and .3;1?. Systems? p ?5/ are all prime.

For example, the obvious map Z! Z? p ?5?=.3;1?. p ?5/ is surjective with kernel .3/, and so. Z? p ?5?=.3;1?. which is an integral domain. How far is this from what we want, namely, unique factorization of elements? In other. words, how many ideal elements have we had to add to our real elements to get unique factorization. In a certain sense, only a finite number: we shall see that there exists a finite set S of ideals such that every ideal is of the form a .a/ for some a 2 S and some a 2OK . Better, we shall construct a group I of fractional ideals in which the principal fractional ideals .a/, a 2K, form a subgroup P of finite index.

The index is called the class number hK of K. We shall see that. hK D 1 OK is a principal ideal domain OK is a unique factorization domain. Unlike Z, OK can have infinitely many units. For example, .1C p 2/ is a unit of infinite. order in Z? p 2? W. p 2/m 1 if m 0: In fact Z? p 2? D f?.1C. p 2/m jm 2 Zg, and so. Z? p 2? f?1gffree abelian group of rank 1g: In general, we shall show (unit theorem) that the roots of 1 in K form a finite group .K/, and that. OK .K/Z r (as an abelian group); moreover, we shall find r: One motivation for the development of algebraic number theory was the attempt to prove Fermats last theorem, i.e., when m 3, there are no integer solutions .x;y;z/ to the equation. with all of x;y;z nonzero. WhenmD 3, this can proved by the method of infinite descent, i.e., from one solution, you show that you can construct a smaller solution, which leads to a contradiction3. The proof makes use of the factorization. Y 3 DZ3?X3 D .Z?X/.Z2CXZCX2/; and it was recognized that a stumbling block to proving the theorem for larger m is that no such factorization exists into polynomials with integer coefficients of degree 2. This led people to look at more general factorizations. In a famous incident, the French mathematician Lame? gave a talk at the Paris Academy in 1847 in which he claimed to prove Fermats last theorem using the following ideas. Let p 2 be a prime, and suppose x, y, z are nonzero integers such that.

Write xp D zp?yp D. Y .z? iy/; 0 i p?1; D e2i=p: He then showed how to obtain a smaller solution to the equation, and hence a contradiction. Liouville immediately questioned a step in Lame?s proof in which he assumed that, in order to show that each factor .z ? iy/ is a pth power, it suffices to show that the factors are relatively prime in pairs and Cuture Comparison: Cultures Essays their product is germ layers a pth power. In fact, Lame? couldnt justify his step (Z?? is not always a principal ideal domain), and Fermats last theorem was not proved for Cuture Indian and Chinese Essays almost 150 years. However, shortly after Lame?s embarrassing lecture, Kummer used his results on the arithmetic of the fields Q?? to prove Fermats 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 compute the Galois group only 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 construc- tions we make in this course. The first assumes the reader knows number theory, whereas the second develops the whole subject algorithmically. Cohens book is the more useful as a supplement to this course, but wasnt available when these notes were first written. George Washington American Revolution? 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 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 Cuture Indian, algebraic number theory, con- centrating on forrest gump the efforts to Cuture Comparison: Indian prove Fermats last theorem. Germ Layers? The notes in Narkiewicz 1990 document the origins of most significant results in algebraic number theory. Lemmermeyer 2009, which explains the origins of ideal numbers, and 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 Cuture and Chinese, 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 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. 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 George and the Revolution Essay the ?i . It consists of all polynomials in the ?i with coefficients in A, i.e., elements of the form X. Cuture Comparison: Indian Essays? 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 on McDonalds b. If aD .a1; . ;am/ and bD .b1; . Cultures? ;bn/, then aCbD .a1; . ;am;b1; . ;bn/: Given an ideal a in A, we can form the quotient ring A=a. Let f WA!

A=a be the homomorphism a 7! aCa; then b 7! f ?1.b/ defines a one-to-one correspondence between the ideals of A=a and the ideals of A containing a, and. 1See also the notes A Primer of Commutative Algebra available on my website. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. A proper ideal a of A is prime if ab 2 a) a or b 2 a. An ideal a is prime if and only if the quotient ring A=a is an integral domain. A nonzero element of A is said to be prime if ./ is a prime ideal; equivalently, if jab) ja or jb. An ideal m in A is maximal if it is germ layers maximal among the proper ideals of A, i.e., if mA 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 Cuture Indian, a maximal ideal if A is Noetherian (see below) this is Quality Systems Essay obvious; otherwise the proof requires Zorns lemma.

In particular, every nonunit in A is Cuture Comparison: Indian and Chinese Cultures Essays contained in a maximal ideal. Forrest Synopsis? 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). Comparison: Indian Cultures Essays? Ideals in products of rings. PROPOSITION 1.1 Consider a product of rings AB . On McDonalds? 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. 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. Cuture Comparison: Indian And Chinese Cultures? Let c be an ideal in AB , and let. aD fa 2 A j .a;0/ 2 cg; bD fb 2 B j .0;b/ 2 cg: Clearly a b c. Conversely, let .a;b/ 2 c. Then .a;0/ D .a;b/ .1;0/ 2 c and .0;b/ D .a;b/ .0;1/ 2 c, and so .a;b/ 2 ab: Recall that an ideal c C is prime if and only if C=c is an integral domain. The map. has kernel ab, and hence induces an isomorphism. Now use that a product of Management Essay, rings is an integral domain if and only if one ring is zero and the other is an integral domain. 2. REMARK 1.2 The lemma extends in an obvious way to a finite product of rings: the ideals in A1 Am are of the form a1 am with ai an ideal in Ai ; moreover, a1 am is prime if and only if there is a j such that aj is a prime ideal in Aj and ai DAi for i j: A ring A is Noetherian if every ideal in A is finitely generated. PROPOSITION 1.3 The following conditions on a ring A are equivalent: (a) A is Noetherian. (b) Every ascending chain of Cuture Indian and Chinese, ideals. eventually becomes constant, i.e., for some n, an 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 series, ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and so an D anC1 D D a. (b) (c): Let a1 2 S . If a1 is not a maximal element of S , then there exists an Cuture Indian and Chinese Cultures, a2 2 S such that a1 a2. Germ Layers? 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 Cuture Comparison: Cultures, A, and let S be the set of finitely generated ideals contained in Quality Systems, 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 Cuture and Chinese Cultures Essays, a properly containing a0. This contradicts the series, definition of a0. 2. A famous theorem of Cuture Comparison: Indian Cultures Essays, Hilbert states that k?X1; . ;Xn? is Noetherian. In practice, al- most all the rings that arise naturally in algebraic number theory or algebraic geometry are Noetherian, but not all rings are Noetherian. For example, the ring k?X1; : : : ;Xn; : : :? of polynomials in an infinite sequence of symbols is Essay on McDonalds Living not Noetherian because the chain of Cuture and Chinese Cultures Essays, 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. We shall need to use that, for elements a and gump b of an integral domain A, .a/ .b/ bja, with equality if and Cuture Comparison: Indian and Chinese Cultures 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 germ layers 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 Cuture Indian Essays not itself irreducible, and so a D bc with b and c nonunits.

Clearly .b/ .a/, and the ideals cant be equal for otherwise c would be a unit. From the maximality of .a/, we deduce that b can be written as a product of forrest gump, irreducible elements, and similarly for c. Thus a is a product of irreducible elements, and we have a contradiction. 2. REMARK 1.5 Note that the proposition fails for the ring O of all algebraic integers in the algebraic closure of Q in C, because, for example, we can keep in extracting square roots an algebraic integer ? can not be an irreducible element of O because. p ? will also be. an algebraic integer and ? D p ? p ?. Indian Cultures? Thus O is not Noetherian. 1. Management Systems Essay? 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 the proof of Proposition 1.3. 2. PROPOSITION 1.7 Let M be an Comparison: Indian Cultures, A-module, and let N be a submodule of M . If N and M=N are both Noetherian, then so also is M . PROOF. I claim that if M 0 M 00 are submodules of M such that M 0N DM 00N and M 0 and M 00 have the same image in on McDonalds vs. Consumer Living, M=N , then M 0 DM 00. To see this, let x 2M 00; the second condition implies that there exists a y 2M 0 with the same image as x inM=N , i.e., such that x?y 2N . Then x?y 2M 00N M 0, and so x 2M 0. Now consider an ascending chain of submodules of M . If M=N is Noetherian, the image of the chain in M=N becomes constant, and if N is Noetherian, the intersection of the chain with N becomes constant. Now the claim shows that the chain itself becomes constant. 2. PROPOSITION 1.8 Let A be a Noetherian ring.

Then every finitely generated A-module is Noetherian. PROOF. If M is generated by a single element, then M A=a for some ideal a in A, and the statement is obvious. Cuture Comparison: Indian? 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 Quality generated by a single element, the statement follows from (1.7). 2. A ring A is said to local if it has exactly one maximal ideal m. In this case, A D Arm (complement of m in A). LEMMA 1.9 (NAKAYAMAS 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 . From the and Chinese Cultures Essays, assumption, we can write. aimi , with ai 2 a, mi 2M: and so mCN 2 a.M=N/: 2. Forrest? 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 Cuture any field M containing A but M 0. Rings of fractions.

Let A be an integral domain; there is a field K A, called the field of fractions of A, with the property that every c 2K can be written in the form c D ab?1 with a;b 2A and b 0. For example, Q is the field of fractions of 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 Living 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. Cuture Indian And Chinese Cultures? For example, if d is a nonzero integer, then2 Zd consists of those elements of Q whose denominator divides some power of d : Zd D fa=dn 2Q j a 2 Z, n 0g: (b) If p is a prime ideal, then SpDArp is a multiplicative set (if neither a nor b belongs to p, then ab does not belong to p/. We write Ap for S?1p A. For example, Z.p/ D fm=n 2Q j n is not divisible by pg:

2This notation conflicts with a later notation in which Zp denotes the ring of 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 georgia series, 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. Indian And Chinese Cultures Essays? Clearly .aA/e a because aA a and a is an ideal in S?1A. For the reverse inclusion, let b 2 a. We can write it b D a=s with a 2 A, s 2 S . Then aD s .a=s/ 2 aA, and and the American so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . Clearly .S?1p/A p. For the reverse inclu- sion, let a=s 2 .S?1p/A, a 2 p, s 2 S . Consider the equation a. s s D a 2 p. Both a=s. and s are in A, and so at least one of a=s or s is in p (because it is prime); but s p (by assumption), and so a=s 2 p: 2. PROPOSITION 1.12 Let A be an integral domain, and let S be a multiplicative subset of A. The map p 7! pe defD p S?1A is a bijection from the set of prime ideals in A such that pS D? to the set of prime ideals in S?1A; the inverse map is p 7! pA. PROOF. It is easy to see that. p a prime ideal disjoint from S) pe is a prime ideal in S?1A, p a prime ideal in S?1A) pA is a prime ideal in A disjoint from S; and (1.11) shows that the two maps are inverse.

2. EXAMPLE 1.13 (a) If p is Cuture Comparison: Indian and Chinese Cultures Essays a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). (b) We list the germ layers, 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 Comparison: Indian Cultures Essays, form xCmd with m 2 Z and d D. We want to translate this in terms of ideals. Integersm and forrest gump n are relatively prime if and only if .m;n/D Z, i.e., if and only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to be relatively prime if aCbD A. If m1; . ;mk are integers, then T .mi / D .m/ where m is the Cuture, least common multiple. of the mi . Thus T .mi / . Q mi /, which equals. Q .mi /. If the mi are relatively prime in. pairs, then mD Q mi , and so we have. Q .mi /. Note that in general, a1 a2 an a1a2 . an; but the two ideals need not be equal. These remarks suggest the following statement. THEOREM 1.14 Let a1; . ;an be ideals in a ring A, relatively prime in pairs. Then for any elements x1; . ;xn of A, the tablets, congruences. have a simultaneous solution x 2 A; moreover, if x is one solution, then the other solutions are the elements of the form xC a with a 2. Q ai . In other words, the. natural maps give an exact sequence.

PROOF. Suppose first that n D 2. As a1C a2 D A, there are elements ai 2 ai such that a1Ca2 D 1. The element x D a1x2Ca2x1 has the required property. 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 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 Cuture Comparison: and Chinese Essays let a1Ca2 D 1, as before. For c 2 a1a2, we have. c D a1cCa2c 2 a1 a2. Aloe? which proves that a1 a2 D a1a2.

We complete the Cuture Essays, proof by induction. This allows us to assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . Forrest Synopsis? The theorem extends to A-modules. THEOREM 1.15 Let a1; . ;an be ideals in A, relatively prime in pairs, and let M be an A-module. There is an exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to use tensor products, which I now review. Review of tensor products. Let M , N , and P be A-modules. A mapping f WM N ! P is Cuture Comparison: Cultures Essays said to be A-bilinear if. Forrest Gump? f .mCm0;n/D f .m;n/Cf .m0;n/ f .m;nCn0/D f .m;n/Cf .m;n0/ f .am;n/D af .m;n/D f .m;an/ 9=; all a 2 A; m;m0 2M; n;n0 2N: i.e., if it is linear in 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 Cuture Indian and Chinese Essays, obvious diagram commute). Tablets? We denote it by 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 Indian Cultures Essays, following two conditions: (a) The mapM N !M ?AN is A-bilinear, and any other A-bilinear mapM N ! P is of the form .m;n/ 7! ?.m?n/ for a unique A-linear map ?WM ?AN ! P ; thus. BilinA.M N;P /D HomA.M ?AN;P /: (b) TheA-moduleM?AN has as generators them?n,m2M , n2N , and as relations. 9=; all a 2 A; m;m0 2M; n;n0 2N: Tensor products commute with direct sums: there is a canonical isomorphism. Review of tensor products.

It follows that if M and N are free A-modules3 with bases .ei / and .fj / respectively, then M ?AN is a free A-module with basis .ei ? fj /. 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 B is r s (so a linear map ks! kr ), then A?B is the mr ns matrix (linear map kns! kmr ) with. 0B@ a11B a1nB. : : : . am1B amnB. 1CA : LEMMA 1.17 If ?WM !M 0 and ?WN !N 0 are surjective, then so also is. ???WM ?AN !M 0 ?AN. PROOF. Recall that M 0?N 0 is 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 germ layers 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 equal to it.

2. One can also show that if M 0!M !M 00! 0. is exact, then so also is. M 0?AP !M ?AP !M 00 ?AP ! 0: For example, if we tensor the exact sequence. with M , we obtain an exact sequence. a?AM !M ! .A=a/?AM ! 0 (2) 3Let M be an A-module. Elements e1; : : : ; em form a basis for M if every element of M can be expressed uniquely as a linear combination of the ei s with coefficients in A. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of A-modules, and M is said to be a free A-module of rank m. 4Kronecker products of matrices pre-date tensor products by about 70 years.

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 and Chinese Cultures Essays 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 a B-module N , there is 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/. 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. Georgia Nicolson Series? 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 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. Cuture Indian Cultures Essays? Therefore f .X/ factors in ??X? into monic irreducible polynomials. Quality Management Systems Essay? that are relatively prime in pairs. We can apply the Chinese Remainder Theorem to deduce that. Finally, ??X? .X// is Indian and Chinese Cultures a finite separable field extension of ? of degree degfi . Germ Layers? Thus we have proved the Cuture Comparison: and Chinese Cultures Essays, following result: THEOREM 1.18 Let K be a finite separable field extension of k, and let ? be an arbitrary field extension. Then K?k? is a product of finite separable field extensions of ?, If ? is a primitive element for K=k, then the image ?i of Quality Management, ? in ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the minimum polynomials for ? and ?i respectively, then. Cuture Indian Cultures Essays? EXAMPLE 1.19 Let K DQ??? with ? algebraic over Q. Then. C?QK ' C?Q .Q?X?=.f .X///' C?X?=..f .X//' Yr. iD1 C?X?=.X ??i / Cr : Here ?1; : : : ;?r are the conjugates of ? in C. The composite of ? 7! 1??WK!

C?QK with projection onto the i th factor is. We note that it is essential to assume in (1.18) that K is separable over k. If not, there will be an ? 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 cant be a product of fields. NOTES Ideals were introduced and studied by Dedekind for rings of algebraic integers, and later by others in polynomial rings. 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. Washington And The American Revolution? If you don't receive any email, please check your Junk Mail box.

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