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How to Write a Bibliography � Examples in MLA Style. Please note, all entries should be typed double-spaced. Saving Private Length! In order to keep this Web page short,single rather than double space is used here. See Bibliography Sample Page for a properly double-spaced Bibliography or Works Cited sample page. Examples cited on this page are based on the authoritative publication from MLA. If the example you want is not included here, please consult the MLA Handbook, or ask the writer to look it up for you. Format for entries: A single space is used after any punctuation mark. By Wilkie Collins! When dividing a long word or URL onto two lines, put hyphen, slash, or period at the end of the line. Do not add a hyphen to a URL that was not originally there. Never begin a new line with a punctuation mark. Double-space all lines in a bibliography entry.

Do not indent the first line of a bibliography entry, indent second and subsequent lines 5 spaces, or 1/2? (1.25 cm) from the left margin. Please see Chapter 11. Guidelines on How to Write a Bibliography for details. When writing a bibliography, remember that the purpose is to ryan communicate to Investigation the reader, in a standardized manner, the sources that you have used in sufficient detail to saving be identified. If you are unable to how technology positively find all the necessary information, just cite what you can find. Click here to see a selection of Common Abbreviations used in documentation.

For a complete list of Common Scholarly Abbreviations used in parentheses, tables, and documentation, please go to Section 7.4 of the 6th edition of the MLA Handbook. Bell, Stewart. The Martyr�s Oath: The Apprenticeship of a Homegrown Terrorist . Mississauga, ON: Wiley, 2005. Biale, David, ed. Cultures of the Jews: A New History . New York: Schocken, 2002. Bowker, Michael. Fatal Deception: The Untold Story of private ryan length Asbestos: Why It Is Still Legal. and Still Killing Us . N.p.: Rodale, 2003. N.p. = No place of publication indicated. Capodiferro, Alessandra, ed.

Wonders of the World: Masterpieces of Architecture from. 4000 BC to the Present . Vercelli: White Star, 2004. Cross, Charles R. Room Full of Mirrors: A Biography of Jimi Hendrix . New York: Maltin, Leonard, ed. Movie Video Guide 2002 Edition . New York: New American, 2001. Meidenbauer, Jorg, ed. Discoveries and Inventions: From Prehistoric to Modern Times . Lisse: Rebo, 2004. Puzo, Mario.

The Family: A Novel . Completed by Carol Gino. New York: Harper, 2001. Rowling, J.K. Harry Potter and Children's Education Investigation the Chamber of saving private ryan Secrets . How Technology! New York: Scholastic, 1999. �. Harry Potter and the Prisoner of Azkaban . Thorndike, ME: Thorndike, 2000. Suskind, Ron. The Price of Loyalty: George W. Bush, the White House, and the Education of.

Paul O�Neill . New York: Simon, 2004. If your citation is from private length one volume of anxiety a multivolume work and each volume has its own title, you need cite only the actual volume you have used without reference to saving other volumes in the work. Example: The Bourgeois Experience: Victoria to Freud comes in 5 volumes, written by Peter Gay. (Title of Vol. 1: Education of the for dummies, Senses ) Gay, Peter.

Education of the Senses . New York: Norton, 1999. (Title of Vol. 2: The Tender Passion) Gay, Peter. Ryan! The Tender Passion . New York: Oxford UP, 1986. (Title of Vol.

3: The Cultivation of Hatred ) Gay, Peter. The Cultivation of Hatred . London: Harper, 1994. (Title of Vol. 4: The Naked Heart ) Gay, Peter. The Naked Heart . New York: Norton, 1995. (Title of how technology has changed positively Vol. Private Length! 5: Pleasure Wars ) Gay, Peter. Pleasure Wars . New York: Norton, 1998. 2. Book with two authors or editors: Bohlman, Herbert M., and Mary Jane Dundas. Collins! The Legal, Ethical and International.

Environment of Business . 5th ed. Cincinnati, OH: West, 2002. Bolman, Lee G., and Terrence E. Deal. Leading with Soul: An Uncommon Journey. of Spirit . Rev. ed. San Francisco: Jossey-Bass, 2001.

Calvesi, Maurizio, and Lorenzo Canova, eds. Rejoice! 700 Years of Art for the Papal. Jubilee . Saving Private Ryan Length! New York: Rizzoli, 1999. Cohen, Andrew, and J.L. Granatstein, eds. Trudeau�s Shadow: The Life and Legacy.

of Pierre Elliott Trudeau . Toronto: Random, 1998. Heath, Joseph, and Andrew Potter. The Rebel Sell: Why the Culture Can�t Be Jammed . 2nd ed. Is Mock! Toronto: Harper, 2005. Llewellyn, Marc, and Lee Mylne. Frommer�s Australia 2005 . Hoboken, NJ: Wiley, 2005. Summers, Anthony, and Robbyn Swan. Sinatra: The Life . Private Ryan! New York: Knopf, 2005. Book prepared for publication by two editors:

Shakespeare, William. The Tragedy of Hamlet, Prince of Denmark . Ed. Barbara A. Mowat and Paul Werstine. New York: Washington. 3. Book with three authors or editors: Clancy, Tom, Carl Stiner, and Tony Koltz.

Shadow Warriors: Inside the Special. Forces . New York: Putnam, 2002. Hewitt, Les, Andrew Hewitt, and Luc d�Abadie. Anxiety! The Power of Focus for College. Students . Deerfield Beach, FL: Health Communications, 2005. Larsson, Mans O., Alexander Z. Saving Ryan! Speier, and Jennifer R. Weiss, eds. Let�s Go: Germany 1998 . New York: St. Martin�s, 1998. Palmer, R.R., Joel Colton, and Lloyd Kramer. A History of the Modern World: To 1815 . 9th ed.

New York: Knopf, 2002. Suzuki, David, Amanda McConnell, and Maria DeCambra. The Sacred Balance: A Visual Celebration of what Our Place in Nature . Vancouver: Greystone, 2002. 4. Book with more than three authors or editors: You have a choice of listing all of the authors or editors in the order as they appear on the title page of the length, book, or use �et al.� from the Latin et alii, or et aliae , meaning �and others� after the first author or editor named. Nelson, Miriam E., Kristin R. Baker, Ronenn Roubenoff, and Lawrence Lindner. Strong Women and Men Beat Arthritis . How Technology! New York: Perigee, 2003. Nelson, Miriam E., et al. Strong Women and Men Beat Arthritis . New York: Hogan, David J., et al., eds.

The Holocaust Chronicle: A History in Words and Pictures . Lincolnwood, IL: International, 2000. Pound, Richard W., Richard Dionne, Jay Myers, and James Musson, eds. Canadian. Facts and Dates . 3rd ed. Markham, ON: Fitzhenry, 2005. Pound, Richard W., et al., eds. Canadian Facts and Dates . 3rd ed.

Markham, ON: Rogerson, Holly Deemer, et al. Words for Students of English: A Vocabulary. Series for ESL . Vol. 6. Private Ryan Length! Advanced Level ESL. Pittsburgh, PA: U of Pittsburgh P, 1989. 5. Book with compilers, or compilers and editors:

McClay, John B., and anxiety Wendy L. Matthews, comps. and eds. Corpus Juris Humorous: A Compilation of Outrageous, Unusual, Infamous and saving Witty Judicial Opinions. from 1256 A.D. to the Present . New York: Barnes, 1994. O�Reilly, James, Larry Habegger, and Sean O�Reilly, comps. and eds. Danger: True Stories of Trouble and Survival . San Francisco: Travellers� Tales, 1999.

Teresa, Mother. It Takes To Break A Man! The Joy in Loving: A Guide to Daily Living with Mother Teresa . Comp. Saving Private Length! Jaya Chaliha and Edward Le Joly. New York: Viking, 1997. Note abbreviation: comp. = compiler or compiled by. 6. Book with no author or editor stated: Maclean�s Canada�s Century: An Illustrated History of the People and Events. That Shaped Our Identity . Toronto: Key, 1999. Microsoft PowerPoint Version 2002 Step by Step . Redmond, WA: Perspection, 2001.

The Movie Book . London: Phaidon, 1999. With Scott to the Pole: The Terra Nova Expedition 1910-1913 . Photographs of. Herbert Ponting. Children's Investigation! New York: BCL, 2004. 7. Book with one author, translated by another: Muller, Melissa. Anne Frank: The Biography . Trans.

Rita and Robert Kimber. New York: Metropolitan, 1998. 8. Saving Ryan Length! Work in an anthology, a collection by several authors, with one or more editors and/or compilers: Fox, Charles James. �Liberty Is Order, Liberty Is Strength.� What Is a Man? 3,000 Years of Wisdom on the Art of Manly Virtue. How Technology! Ed. Waller R. Newell. New York: Harper, 2001. 306-7. Wilcox, Robert K. Saving Private Length! �Flying Blind.� Danger: True Stories of Trouble and Survival . Comp. and ed.

James O�Reilly, Larry Habegger, and Sean O�Reilly. San Francisco: Travellers� Tales, 1999. For Dummies! 211-22. 9. Saving! Article in an encyclopedia with no author stated: �Nazi Party.� New Encyclopaedia Britannica . 1997 ed. �Tajikistan.� World Book Encyclopedia of People and Places . 2000 ed. 10. Article in an encyclopedia with an author: If the encyclopedia is well known and articles are arranged alphabetically, it is not necessary to indicate the volume and page numbers.

If the encyclopedia is not well known, you must give full publication information including author, title of article, title of encyclopedia, name of editor or edition, number of volumes in the set, place of publication, publisher and year of publication. Kibby, Michael W. The Moonstone! �Dyslexia.� World Book Encyclopedia . 2000 ed. Midge, T. �Powwows.� Encyclopedia of saving length North American Indians . Education! Ed. D.L. Birchfield. 11 vols. Private Ryan Length! New York: Cavendish, 1997. 11.

Article in a magazine, journal, periodical, newsletter, or newspaper with no author stated: �100 Years of Dust and Glory.� Popular Mechanics Sept. 2001: 70-75. �Celestica to Repair Palm Handhelds.� Globe and Mail [Toronto] 29 Oct. How Technology Has Changed Positively! 2002: B6. �E-Money Slips Quietly into Oblivion.� Nikkei Weekly [Tokyo] 22 Jan.

2001: 4. �McDonald�s Declines to Fund Obesity Education on Danger of Eating Its Food.� National Post [Toronto] 18 Apr. 2006: FP18. �Pot Use Doubled in Decade, Study Says: 14% Smoked Up in the Past Year.� Toronto Star. 25 Nov. 2004: A18.

�Secondhand Smoke Reduces Kids� IQs.� Buffalo News 23 Jan. Private Ryan Length! 2005: I6. 12. Article in a magazine, journal, periodical, newsletter, or newspaper with one or more authors: Use �+� for pages that are not consecutive. Example: When numbering pages, use �38-45� if page numbers are consecutive. Use �A1+� if article begins on 50 mg anxiety, page A1, contains more than one page, but paging is not consecutive. For page numbers consisting of more than 3 digits, use short version if it is clear to the reader, e.g.

220-268 may be written as 220-68, but 349-560 must be written in full. Note also that there is no period after the month. The period in �Mar.� is for the abbreviation of March. If there are 4 or less letters in the month, e.g. May, June, and saving private July, the months are not abbreviated. If the publication date is July 18, 2005, citation will be 18 July 2005. Where a journal or magazine is a weekly publication, �date, month, year� are required. Where a journal or magazine is a monthly publication, only �month, year� are needed. Where a newspaper title does not indicate the location of publication, add the city of publication between square brackets, e.g. Daily Telegraph [London].

Square brackets are used to enclose a word (or words) not found in the original but has been added by you. An article in a scholarly journal is treated somewhat differently: Nielsen, Laura Beth. �Subtle, Pervasive, Harmful: Racist and Sexist Remarks in. Public as Hate Speech.� Journal of Social Issues 58.2 (2002): 265. The above citation shows: Author�s name, Article title, Name of scholarly journal (underlined), Volume number, Issue number, Year of publication (in parentheses), and Page number. By Wilkie! If the saving private ryan, article is accessed online, add Access date and by wilkie URL at the end, see 23. Internet citations, or citing electronic sources (e). Bogomolny, Laura. �Boss Your Career.� Canadian Business 13-16 Mar.

2006: 47-49. Cave, Andrew. Saving Private Length! �Microsoft and what Sun Settle Java Battle.� Daily Telegraph [London] Cohen, Stephen S., and J. Bradford DeLong. �Shaken and Stirred.� Atlantic Monthly. Jan.-Feb. 2005: 112+. Coleman, Isobel. �Women, Islam, and the New Iraq.� Foreign Affairs Jan.-Feb. 2006: 24+. Daly, Rita. Length! �Bird Flu Targeting the is mock, Young.� Toronto Star 11 Mar. Saving! 2006: A1+. Dareini, Ali Akbar. A Lot To Break A Man Essay! �Iranian President Defends Country�s Nuclear Ambitions.� Buffalo News. Hewitt, Ben. �Quick Fixes for Everyday Disasters.� Popular Mechanics Nov.

2004: 83-88. Johnson, Linda A. �Fight Flu with Good, Old Advice from Mom.� Buffalo News. 10 Oct. 2004: A1-2. Mather, Victoria. �In Tiger Country.� Photos by James Merrell. Town Country Travel. Fall 2004: 102-111. Mohanty, Subhanjoy, and Ray Jayawardhana. �The Mystery of Brown Dwarf Origins.� Scientific American Jan.

2006: 38-45. Petroski, Henry. �Framing Hypothesis: A Cautionary Tale.� American Scientist Jan.-Feb. Plungis, Jeff, Ed Garsten, and Mark Truby. �Caremakers� Challenge: Green, Mean. Machines.� Detroit News and Free Press Metro ed. Saving Ryan Length! 12 Jan. 5 Ways! 2003: 1A+. Sachs, Jeffrey D. �A Practical Plan to End Extreme Poverty.� Buffalo News 23 Jan. 2005: I2. Saletan, William. �Junk-Food Jihad.� National Post [Toronto] 18 Apr. 2006: A18.

Thomas, Cathy Booth, and private Tim Padgett. To Break! �Life Among the Ruins.� Time 19 Sept. 2005: 28+. Wolanski, Eric, Robert Richmond, Laurence McCook, and Hugh Sweatman. �Mud, Marine Snow and Coral Reefs.� American Scientist Jan.-Feb. 2003: 44-51. Wolanski, Eric, et al. Saving! �Mud, Marine Snow and Coral Reefs.� American Scientist. Jan.-Feb. 2003: 44-51.

13. Article from education SIRS (Social Issues Resources Series): Suggested citation example from length SIRS: Bluestone, Barry, and Irving Bluestone. �Workers (and Managers) of the World Unite.� Technology Review Nov.-Dec. 1992: 30-40. Reprinted in WORK . (Boca Raton, FL: Social Issues Resource Series, 1992), Article No. 20. Bluestone, Barry, and Irving Bluestone. What Is Mock! �Workers (and Managers) of the World Unite.� Technology Review Nov.-Dec.

1992: 30-40. Work . Ed. Eleanor Goldstein. Vol. 5. Boca Raton: SIRS, 1992. Art. 20. Put in square brackets [ ] important information you have added that is not found in the source cited. Build-a-Bear. Saving Private! Advertisement.

7 Feb. 2005 http://www.buildabear.com/shop/default.aspx. GEICO. Advertisement. Newsweek 16 Jan. 2006: 92. IBM. Advertisement. The Moonstone By Wilkie! Globe and Mail [Toronto] . 29 Oct. 2002: B7. Toyota. Advertisement.

Atlantic Monthly . Private! Jan.-Feb. 2005: 27-30. 15. What Epic! Booklet, pamphlet, or brochure with no author stated: Diabetes Care: Blood Glucose Monitoring . Saving Private Length! Burnaby, BC: LifeScan Canada, 1997. 16.

Booklet, pamphlet, or brochure with an author: Zimmer, Henry B. Canplan: Your Canadian Financial Planning Software . Calgary, AB: May use short forms: Rev. (Review), Ed. (Edition, Editor, or Edited), Comp. (Compiled, Compiler). Creager, Angela N.H. �Crystallizing a Life in Science.� Rev. of Rosalind Franklin: The. Dark Lady of DNA , by Brenda Maddox. American Scientist Jan.-Feb. 2003: 64-66. Dillon, Brenda. �Hana�s Suitcase.� Rev. of Hana�s Suitcase , by Karen Levine. Professionally Speaking June 2003: 36.

Foley, Margaret. �Measured Deception.� Rev. of The Measure of All Things: The. Seven-Year Odyssey and what Hidden Error That Transformed the World, by saving private length, Ken Alder. Discover Nov. 2002: 77. Groskop, Viv. �Chinese Torture � at Five.� Rev. of The Binding Chair, by Kathryn. Harrison. International Express 6 June 2000, Canadian ed.: 37. Hoffman, Michael J. �Huck�s Ironic Circle.� Rev. of The Adventures of Huckleberry.

Finn , by Mark Twain. What Is Mock Epic! Modern Critical Interpretations of Mark Twain�s. Adventures of Huckleberry Finn, ed. Harold Bloom. Saving Private! New York: Chelsea, Iragui, Vicente.

Rev. of Injured Brains of Medical Minds: Views from Within , comp. and ed. Narinder Kapur. New England Journal of Medicine 26 Feb. 1998: Neier, Aryeh. �Hero.� Rev. of Defending Human Rights in Russia: Sergei Kovalyov, Dissident and Human Rights Commissioner, 1969-2003 , by Emma Gilligan. New York Review of Books 13 Jan. 2005: 30-33. Onstad, Katrina. �A Life of Pain and Paint.� Rev. of Frida , dir.

Julie Taymor. National. Post [Toronto] 1 Nov. 2002: PM1+. Redekop, Magdalene. �The Importance of Being Mennonite.� Rev. of A Complicated. Kindness, by Miriam Toews. Children's Education! Literary Review of Canada Oct. 2004: 19-20.

Simic, Charles. �The Image Hunter.� Rev. of Joseph Cornell: Master of Dreams , by. Diane Waldman. New York Review 24 Oct. 2002: 14+. 18. CD-ROM, DVD: See also 35.

Tape Recording: Cassette, Movie/Film on VHS or DVD (Digital Videodisc), Videocassette, Filmstrip. A Place in the Sun . Saving Ryan Length! Dir. George Stevens. 1951. DVD. Paramount, 2001 . Encarta 2004 Reference Library . CD-ROM. Microsoft, 2003 . Encarta 2004 Reference Library Win32 . Educ. ed. DVD. Microsoft, 2003.

LeBlanc, Susan, and Cameron MacKeen. �Racism and atenolol 50 mg anxiety the Landfill.� Chronicle-Herald. 7 Mar. Length! 1992: B1. CD-ROM. SIRS 1993 Ethnic Groups. Vol. 4. Art. 42. Links 2003: Championship Courses . CD-ROM. Microsoft Game Studios, 2002.

YellowPages.city: Toronto-Central West Edition , 1998. CD-ROM. Montreal: 19. Computer service � e.g. BRS, DIALOG, MEAD, etc.: Landler, Mark. �Can U.S. The Moonstone By Wilkie! Companies Even Get a Bonjour?� New York Times , Late Ed. Ryan Length! � Final Ed., 1. 2 Oct. 1995.

DIALOG File 472, item 03072065. When citing a definition from a dictionary, add the abbreviation Def. after the word. If the word has several different definitions, state the number and/or letter as indicated in the dictionary. �Mug.� Def. 2. The New Lexicon Webster�s Encyclopedic Dictionary of the. English Language . Children's! Canadian ed. Saving Private Ryan Length! 1988. Short forms may be used, e.g. dir. (directed by), narr. Is Mock! (narrated by), perf. (performers), prod. (produced by), writ. (written by).

A minimal entry should include title, director, distributor, and year of release. Saving! May add other information as deemed pertinent between the title and what is mock epic the distributor. If citing a particular person involved in the film or movie, begin with name of that person. Charlie and the Chocolate Factory . Saving Private Ryan Length! Dir. Tim Burton. Based on 50 mg anxiety, book by Roald Dahl.

Perf. Johnny Depp. Warner, 2005. Depp, Johnny, perf. Charlie and the Chocolate Factory . Dir. Tim Burton. Based on book. by Roald Dahl. Warner, 2005. Burton, Tim, dir.

Charlie and the Chocolate Factory . Based on book by Roald Dahl. Perf. Johnny Depp. Warner, 2005. Monster-in-Law . Saving Private Length! Dir. Robert Luketic. Writ. Anya Kochoff. Prod. The Moonstone! Paula Weinstein,

Chris Bender, and J.C. Spink. Perf. Saving Private Length! Jennifer Lopez and Children's Education Jane Fonda. New Line, 2005. Nanny McPhee . Private! Dir. Kirk Jones. Based on Nurse Matilda Books Writ. Christianna. Brand. Prod.

Lindsay Doran, Tim Bevan, and is mock epic Eric Fellner. Perf. Emma Thompson, Colin Firth, and Angela Lansbury. Universal, 2005. One Hour Photo . Writ. and dir. Mark Romanek. Prod. Christine Vachon, Pam Koffler, and Stan Wlodkowski. Perf.

Robin Williams. Fox Searchlight, 2002. Titanic . Dir., writ., prod., ed. James Cameron. Prod. Jon Landau. Twentieth. Century Fox and saving private length Paramount, 1997. The Tuxedo . Dir. A Lot To Break A Man! Kevin Donovan. Prod.

John H. Williams, and Adam Schroeder. Perf. Jackie Chan and Jennifer Love Hewitt. DreamWorks, 2002. Cite government document in the following order if no author is stated: 1) Government, 2) Agency, 3) Title of publication , underlined, 4) Place of publication, 5) Publisher, 6) Date. Canada. Minister of Indian Affairs and saving length Northern Development.

Gathering Strength: Canada�s Aboriginal Action Plan . Positively! Ottawa: Minister of Public Works and. Government Services Canada, 2000. United States. Saving Ryan Length! National Council on by wilkie collins, Disability.

Carrying on the Good Fight � Summary Paper from Think Tank 2000 � Advancing the Civil and ryan length Human. Rights of People with Disabilities from aquinas 5 ways Diverse Cultures . Saving! Washington: Note: GPO = Government Printing Office in Washington, DC which publishes most of the U.S. federal government documents. In citing a Congressional Record, abbreviate and underline the term, skip all the details and indicate only the date and is mock page numbers. Example � for the following record:

United States. Personal Responsibility and private length Work Opportunity Reconciliation Act of 1996 . PL 104-193. By Wilkie Collins! Congressional Record. Washington: GPO, July 31, 1996. Cong. Rec . 31 July 1996: 104-193. For examples on private ryan length, how to how technology positively cite more complicated government documents, please see Section 5.6.21 in private, MLA Handbook for Children's Writers of Research Papers, 6th ed. 23. Internet citations, or citing electronic sources:

Basic components of an Internet citation: 2) �Title of Article, Web page or site� in quotation marks. 3) Title of Magazine, Journal, Newspaper, Newsletter, Book, Encyclopedia, or Project , underlined. 5) Indicate type of material, e.g. Saving! advertisement, cartoon, clipart, electronic card, interview, map, online posting, photograph, working paper, etc. if not obvious. 6) Date of has changed positively article, of Web page or site creation, revision, posting, last update, or date last modified. 7) Group, association, name of forum, sponsor responsible for Web page or Web site. 8) Access date (the date you accessed the length, Web page or site). 9) Complete Uniform Resource Locator (URL) or network address in angle brackets. Note: An exception is 50 mg made in referencing a personal e-mail message where an individual�s e-mail address is omitted for saving private length privacy reasons. Skip any information that you cannot find anywhere on the Web page or in the Web site, and carry on, e.g. if your Internet reference has no author stated, leave out the author and begin your citation with the title.

Always put your access date just before the URL which is placed between angle brackets or �less than� and �greater than� signs at the end of the citation. Generally, a minimum of three items are required for an Internet citation: Title, Access Date, and URL. If the URL is too long for a line, divide the address where it creates the what epic, least ambiguity and confusion, e.g. do not divide a domain name and length end with a period such as geocities . Do not divide a term in It Takes a Man Essay, the URL that is saving ryan made up of combined words e.g. SchoolHouseRock . Never add a hyphen at for dummies the end of the line to indicate syllabical word division unless the hyphen is actually found in the original URL. Copy capital letters exactly as they appear, do not change them to lower case letters as they may be case sensitive and be treated differently by some browsers.

Remember that the purpose of indicating the URL is for readers to private length be able to access the Web page. It Takes To Break Essay! Accuracy and private ryan length clarity are essential. a. Internet citation for an advertisement: IBM. Advertisement. 23 Mar. 2003 http://www.bharatiyahockey.org/2000Olympics/ TheraTears. Advertisement. 2003.

8 May 2004 http://www.theratears.com/dryeye.htm. b. Internet citation for an article from an online database (e.g. SIRS, eLibrary), study guide, magazine, journal, periodical, newsletter, newspaper, online library subscription database service, or an article in by wilkie, PDF with one or more authors stated: Bezlova, Antoaneta. �China to Formalize One-Child Policy.� Asia Times Online . 24 May 2001. 10 Oct. 2005 http://www.atimes.com/china/CE24Ad02.html. Clifford, Erin. �Review of ryan length Neuropsychology.� SparkNotes . 10 Oct. 2005. Machado, Victoria, and George Kourakos.

IT Offshore Outsourcing Practices in Canada . Ottawa: Public Policy Forum, 2004. 10 Oct. 2005 http://www.ppforum.com/ow/it_outsourcing.pdf. Marshall, Leon. Collins! �Mandela in saving ryan length, Retirement: Peacemaker without Rest.� 9 Feb. 2001. National Geographic 10 Oct. 2005 http://news.nationalgeographic.com/news/ Thomason, Larisa. �HTML Tip: Why Valid Code Matters.� Webmaster Tips. Newsletter . Dec.

2003. NetMechanic. 10 Oct. 2005 http://www.netmechanic.com/ If using an online library subscription database service, add the name of the service, the name of the library or library system, plus the location of the Children's, library where the database is accessed, e.g.: Gearan, Anne. Ryan Length! �Justice Dept: Gun Rights Protected.� Washington Post . 8 May 2002. SIRS.

Iona Catholic Secondary School, Mississauga, ON. 23 Apr. 2004. Note: 8 May 2002 = date of publication, 23 Apr. 2004 = date of access. Is Mock! Indicate page numbers after publication date if available, e.g. 8 May 2002: 12-14. Leave out saving ryan length page numbers if not indicated in source. Pahl, Greg. �Heat Your Home with Biodiesel�. Mother Earth News . 12 Jan. Children's Education Investigation! 2003.

eLibrary Canada. Twin Lakes Secondary School, Orillia, ON. 10 Apr. 2006. Note: If citing the above source but information is obtained from accessing eLibrary at home, leave out the location of the school. Pahl, Greg. �Heat Your Home with Biodiesel�.

Mother Earth News . 12 Jan. 2003. eLibrary Canada. 10 Apr. Private Ryan Length! 2006. http://www.proquestk12.com. c. Internet citation for an article from an online encyclopedia:

Duiker, William J. �Ho Chi Minh.� Encarta Online Encyclopedia . 2005. Microsoft. 10 Oct. 2005. �Ho Chi Minh.� Encyclop?dia Britannica . By Wilkie! 2005. Saving Length! Encyclop?dia Britannica Premium Service.

9 Oct. 2005 http://www.britannica.com/eb/article-9040629. �Royal Shakespeare Company (RSC).� Britannica Concise Encyclopedia . Education Investigation! 2005. Encyclop?dia Britannica. 8 Oct. 2005 http://concise.britannica.com/ebc/article?eu=402567. d. Internet citation for an article from an private ryan online magazine, journal, periodical, newsletter, or newspaper with no author stated: �Childcare Industry �Should Welcome Men�.� BBC News Online: Education .7 June 2003.

10 Oct. 2005 http://news.bbc.co.uk/1/low/education/2971310.stm. �Taiwan: A Dragon Economy and the Abacus.� BrookesNews.Com . 50 Mg Anxiety! 8 Dec. Saving Private Length! 2003. 10 Oct. Atenolol! 2005 http://www.brookesnews.com/030812taiwan.html. e. Internet citation for an article in length, a scholarly journal: Nielsen, Laura Beth. �Subtle, Pervasive, Harmful: Racist and by wilkie collins Sexist Remarks in. Public as Hate Speech.� Journal of Social Issues 58.2 (2002), 265-280. 7 June 2003. f. Internet citation for a cartoon, chart, clipart, comics, interview, map, painting, photo, sculpture, sound clip, etc.: �Islamic State of Afghanistan: Political Map.� Map.

Atlapedia Online . 1993-2003. Latimer Clarke. Private Ryan! 7 June 2003 http://www.atlapedia.com/online/maps/ Kersten, Rick, and Pete Kersten. �Congratulations!� Electronic card. Blue Mountain Arts . 2000. 7 June 2003 http://www.bluemountain.com/ Lee , Lawrence. Interview. JournalismJobs.com . Feb. 2003. 10 Oct.

2005. Schulz, Charles. �Peanuts Collection � Snoopy Cuddling Woodstock.� Cartoon. By Wilkie! Art.com . 25 Apr. 2004 http://www.art.com/asp/sp.asp?PD=10037710RFID=814547. �Woodhull, Victoria C.� American History 102 Photo Gallery. 1997.

State. Historical Society of Wisconsin. 10 Oct. 2005 http://us.history.wisc.edu/ g. Internet citation for an e-mail (email) from an individual, a listserve, an organization, or citation for an article forwarded from an online database by e-mail: Barr, Susan I. �The Creatine Quandry.� Bicycling Nov. 1998. EBSCOhost Mailer.

E-mail to E. Interior. 11 May 2003. Kenrick, John. �Re: Link to Musicals101.com.� E-mail to length I. Lee. 10 May 2003. �NEW THIS WEEK for September 8, 2005.� E-mail to author. 8 Sept. 2005.

PicoSearch. �Your PicoSearch Account is Reindexed.� E-mail to John Smith. h. Education! Internet citation for an online government publication: Canada. Office of the Auditor General of Canada and the Treasury Board. Secretariat. Modernizing Accountability Practices in the Public Sector . 6 Jan.

1998. 10 Oct. 2005 http://www.tbs-sct.gc.ca/rma/account/ United States. National Archives and Records Administration. The Bill of Rights . 29 Jan. 1998. 10 Oct. 2005 http://www.archives.gov/exhibit_hall/ i. Internet citation for an online posting, forum, letter to the editor:

Kao, Ivy. �Keep Spreading the Word.� Online posting. 4 June 2003. Reader Responses, Opinion Journal, Wall Street Journal Editorial Page . 10 Oct. 2005. Seaside Harry . �My Friend Drove My Car with the Parking Brake On!� Online. posting. 10 Oct. Ryan Length! 2005. PriusOnline.com Forum Index � Prius � Technical . 10 Oct.

2005 http://www.priusonline.com/viewtopic.php?t=6298highlight=. j. Internet citation for what is mock epic an online project, an information database, a personal or professional Web site: The MAD Scientist Network . 1995-2001 or 30 Feb. 1906. Washington U. School of Medicine.

10 Oct. 2005. http://www.madsci.org. O�Connor, J.J., and E.F. Saving Private! Robertson. �John Wilkins.� Feb. 2002. The Moonstone By Wilkie! U of saving private ryan length St. Andrews, Scotland. 10 Oct. 2005 http://www-history.mcs.st-andrews.ac.uk/history/ Officer, Lawrence H. �Exchange Rate between the United States Dollar and Forty.

Other Countries, 1913 -1999.� Economic History Services, EH.Net, 2002. 13 Apr. 2006 http://www.eh.net/hmit/exchangerates/. Savill, R. Richard. �Jazz Age Biographies.� The Jazz Age Page . 23 Oct. 2000. 12 Apr. 2006 http://www.btinternet.com/ Sullivan, Danny. �Search Engine Math.� 26 Oct. 2001. Search Engine Watch . 10 Apr. 2006 http://www.searchenginewatch.com/facts/math.html.

Wurmser, Meyrav, and Yotam Feldner. �Is Israel Negotiating with the Hamas?� Inquiry and Analysis No. 16. 23 Mar. 1999. The Middle East Media and. Research Institute. 10 Oct.

2005 http://memri.org/bin/articles.cgi? k. Internet citation for a software download: It is not essential to by wilkie collins include the file size. Do so if preferred by your instructor. RAMeSize . Vers. 1.04. 15K. 24 Sept. 2000. Blue Dice Software.

12 Oct. 2004. l. Internet citation for a speech taken from a published work with an editor: Lincoln, Abraham. �The Gettysburg Address.� 19 Nov. 1863. The Collected Works of. Abraham Lincoln . Saving Private Length! Ed. Roy P. Basler. New Brunswick, NJ: Rutgers UP,

1955. Abraham Lincoln Online. 10 Oct. 2005 http://showcase.netins.net/ m. Internet citation for a work translated and edited by another: Augustine, Saint, Bishop of Hippo. Confessions Enchiridion . Trans. and ed. Albert C. Outler. 1955. Dallas, TX: Southern Methodist U. Digitized 1993.

10 Oct. 2005 http://www.ccel.org/a/augustine/confessions/ Blair, Tony. Interview. Prime Minister�s Office. 31 May 2003. 13 Apr. 2006. Chirac, Jacques. Interview. Time 16 Feb. 2003. 10 Oct. 2005. Longin, Hellmut. Telephone interview. 3 May 2006. Neilsen, Jerry. E-mail interview. 28 Apr. 2006. Wyse, Randall. Personal interview. 24 July 2005.

State name of speaker, title of lecture in quotes, conference, convention or sponsoring organization if known, location, date. Bradley, Vicki. �Marriage.� Agnes Arnold Hall, U of Houston. 15 Mar. 2003. Wilson-Smith, Anthony. �Hello, He Must Be Going.� Editorial. Epic! Maclean�s 26 Aug. 2002: 4. Lange, Rick. �U.N. Has Become Ineffective and Ought to Be Disbanded.� Letter. Buffalo. News 23 Jan.

2005: I5. Woods, Brede M. Letter. Newsweek 23 Sept. 2002: 16. Kolbert, Elizabeth. �Six Billion Short: How Will the Mayor Make Ends Meet?� Letter. New Yorker 13 Jan. 2003: 33-37. Geens, Jennifer. Private Length! Reply to how technology education positively letter of Bill Clark. Toronto Star 29 Sept.

2002: A1. A letter you received from John Smith: Smith, John. Letter to the author. 15 June 2005. Twain, Mark. �Banned in Concord.� Letter to Charles L. Webster. 18 Mar. 1885. Letter 850318 of Mark Twain . Ed.

Jim Zwick. Length! 2005. 10 Oct. 2005. Treat citation as if it is a book with no author stated. Indicate if the citation is for to Break a chart or a map. 2004 Andex Chart . Chart. Windsor, ON: Andex, 2004.

Canada . Map. Ottawa: Canadian Geographic, 2003. �Dallas TX.� Map. 2005 Road Atlas: USA, Canada, Mexico . Greenville, SC: Michelin, 2005. Components: 1) Name of composer. 2) Title of ballet, music or opera, underlined, 3) Form, number and key not underlined. Beethoven, Ludwig van. Fur Elise.

Strauss, Richard. Traumerei , op. 9, no. 4. Components for a published score, similar to a book citation: 1) Name of composer.

2) Underline title of private ryan ballet, music, opera, as well as no. and op., important words capitalized, prepositions and conjunctions in what is mock, lower case. 3) Date composition written. 4) Place of publication: 5) Publisher, 6) Date of publication. Chopin, Frederic. Mazurka Op. 7, No. 1 . New York: Fischer, 1918. Ledbetter, Huddie, and private length John Lomax. Goodnight, Irene . 1936. New York: Spencer, 1950.

Stier, Walter C. Education! Sweet Bye and Bye . London: Paxton, 1953. Weber, Carl Maria von. Invitation to the Dance Op. Private Length! 65 . 1819. London: Harris, 1933. 29. Painting, photograph, sculpture, architecture, or other art form. Components for education citing original artwork: 1) Name of artist. 2) Title of saving private ryan artwork, underlined.

3) Date artwork created. 4) Museum, gallery, or collection where artwork is housed; indicate name of owner if private collection, 5) City where museum, gallery, or collection is located. Ashoona, Kiawak. Smiling Family . Education Positively! 1966. McMichael Canadian Art Collection, Brancusi, Constantin. The Kiss . 1909.

Tomb of T. Rachevskaia, Montparnasse. The Great Sphinx . [c. 2500 BC]. Giza. Ingres, Jean-Auguste-Dominique. Odalisque . 1814. Louvre Museum, Paris. Raphael. The School of Athens . 1510-11.

Stanza della Segnatura, Vatican Palace, Rude, Francois. La Marseillaise . Private! 1833-36. The Moonstone By Wilkie! Arc de Triomphe, Paris. Components for artwork cited from a book: 1) Name of artist. 2) Underline title of artwork. 3) Date artwork created (if date is saving ryan uncertain use [c. 1503] meaning [circa 1503] or around the year 1503). 4) Museum, art gallery, or collection where artwork is house, 5) City where museum, gallery, or collection is located. 6) Title of aquinas 5 ways for dummies book used. 7) Author or editor of book.

8) Place of publication: 9) Publisher, 10) Date of publication. 11) Other relevant information, e.g. Saving! figure, page, plate, or slide number. Abell, Sam. Japan . 1984. National Geographic Photographs: The Milestones . By Leah Bendavid-Val, et al.

Washington, DC: National Geographic, 1999. Carr, Emily. A Haida Village . [c. 1929]. McMichael Canadian Art Collection, Kleinburg, ON. The McMichael Canadian Art Collection . How Technology Education! By Jean Blodgett, et al. Toronto: McGraw, 1989. Saving Length! 134.

Kasebier, Gertrude. The Magic Crystal . [c. 1904]. Royal Photographic Society, Bath. Atenolol! A Basic History of Art . By H.W. Janson and Anthony F. Janson. Englewood Cliffs, NJ: Prentice, 1991. 412. Leonardo, da Vinci. Mona Lisa (La Gioconda) . [c.

1503-5]. Louvre Museum, Paris. Favorite Old Master Paintings from the Louvre Museum . New York: Abbeville, 1979. 31. Michelangelo. David . 1501-04. Accademia di Belle Arti, Florence. The Great. Masters . By Giorgio Vasari.

Trans. Gaston Du C. de Vere. New York: Park Lane, 1986. 226. Sullivan, Louis. Wainright Building . 1890-91. St. Louis, MO. A Basic History of Art . By H.W. Janson and Anthony F. Janson.

Englewood Cliffs, NJ: Prentice, Tohaku, Deme. Ko-omote Female Mask . Edo period [1603-1867], Japan. Naprstek. Museum, Prague. The World of Masks . By Erich Herold, et al. Saving Ryan Length! Trans. Dusan. Zbavitel. London: Hamlyn, 1992.

207. Vanvitelli, Luigi, and Nicola Salvi. Chapel of St. John the epic, Baptist . 1742-51. Sao Roque, Lisbon. By Rolf Toman, ed. Baroque: Architecture, Sculpture, Painting . Cologne: Konemann, 1998. 118.

Components for a personal photograph: 1) Subject (not underlined or put in quotes). 2) Name of person who took the photograph. 3) Date photograph taken. War in Iraq: Operation Iraq Freedom on CNN. Ryan! Personal photograph by author. Great Wall of China, Beijing, China. Positively! Personal photograph by Cassy Wyse. 28 July 2005. Components: 1) Patent inventor(s) or owner(s). 2) Title of patent. 3) Issuing country and patent number.

4) Date patent was issued. Arbter, Klaus, and ryan Guo-Qing Wei. �Verfahren zur Nachfuhrung eines Stereo-Laparoskope. in der minimal invasiven Chirurgie.� German Patent 3943917. July 1996. �Conversion of Calcium Compounds into Solid and how technology positively Gaseous Compounds.� US Patent 5078813. Kamen, Dean L., et al. Private Ryan! �Transportation Vehicles and Methods.� US Patent 5971091. 31. Performance: (ballet, concert, musical, opera, play, theatrical performance) Disney�s The Lion King . By Roger Allers and Irene Mecchi.

Dir. Julie Taymor. Music and lyrics by Elton John and Tim Rice. Princess of Wales Theatre, Toronto. 9 June 2002. The Hobbit . By J.R.R. What Is Mock Epic! Tolkien. Saving Private! Dir. Kim Selody. Perf. Herbie Barnes, Michael.

Simpson, and Chris Heyerdahl. Living Arts Centre, Mississauga, ON. The Nutcracker . Education! By Pyotr Ilyich Tchaikovsky. Chor. and Libretto by saving private ryan length, James. Kudelka. Cond. Ormsby Wilkins and Uri Mayer. Has Changed! National Ballet of. Canada.

Hummingbird Centre, Toronto. 30 Dec. 1999. Phantom of the ryan length, Opera . By Andrew Lloyd Webber. A Man! Lyrics by Charles Hart. Dir. Harold Prince. Based on novel by ryan, Gaston Leroux. Pantages Theatre, Toronto.

20 Sept. 1998. The Shanghai Acrobats . By Incredible! Acrobats of China. By Wilkie Collins! Living Arts Centre, Mississauga, ON. 4 Mar. Private Ryan Length! 2005. Components: 1) Title of episode, underlined; or in quotes if appropriate.

2) Title of program, underlined. For Dummies! 3) Title of saving ryan series. 4) Name of network. 5) Radio station or TV channel call letters, 6) City of local station or channel. 6) Broadcast date. The CFRB Morning Show . By Ted Woloshyn. CFRB Radio, Toronto. 12 Sept.

2003. Law and Order . Prod. Wolf Film, Universal Television. NBC Television Network. WHEC, Rochester, NY. 16 Oct. 2002. �New Threat from Osama?� By Jim Stewart. CBS News . WBEN, Buffalo. �New York Museum Celebrates Life of aquinas for dummies Einstein.� By Martha Graybow. Reuters,

New York. WBFO, Buffalo. 13 Nov. Private Length! 2002. �The Nightmare Drug.� By Bob McKeown, Linden MacIntyre, and Hana Gartner. The Fifth Estate . CBC, Toronto. 16 Oct. 2002. �U.S.: Tape Sounds Like Bin Laden.� AP, Washington, DC.

On Your Side . WGRZ-TV, Buffalo. The Moonstone! 13 Nov. 2002. 33. Recording � Music CD, LP, magnetic tape: 1) Name of author, composer, singer, or editor. 2) Title of song (in quotation marks).

3) Title of recording (underlined). 4) Publication medium (LP, CD, magnetic tape, etc.). Saving Private! 5) Edition, release, or version. 6) Place of publication: Publisher, Date of publication. If citing from Internet, see Item 23. Backstreet Boys. Education! Larger than Life . Saving Length! Millennium. CD. Exclusive Management by. The Firm, Los Angeles, CA.

Mastered by Tom Coyne, Sterling Sound, NYC. Burch, Marilyn Reesor. Mosaic . CD. Writ., dir. and prod. Marilyn Reesor. Burch. Choirs dir. Don and Catherine Robertson. Collins! Barrie, ON: Power.

Plant Recording Studio, n.d. Burch, Marilyn Reesor. Saving Ryan! Mosaic . CD. Writ., dir. and prod. Marilyn Reesor. Burch. Choirs dir. Don and Catherine Robertson. Barrie, ON: Power. Plant Recording Studio, [c.

1997]. Note: �n.d.� means �no date� available. What Epic! [c. 1997] means �circa 1997.� McDonald, Michael. No Lookin� Back . LP. Prod. Michael McDonald and. Ted Templeman. Engineered and mixed by R.

ThinkPad ACP Patch for ThinkPad 600, 770, and 770E . Diskette. Vers. 1.0. Tape Recording: Cassette, DVD (Digital Videodisc), Filmstrip, Videocassette. Covey, Stephen R. Living the 7 Habits: Applications and saving private ryan length Insights . Cassette. tape recording read by 50 mg anxiety, author. New York: Simon, Audio Div., 1995. Ginger . Solid Ground. Cassette tape recording from album Far Out . Vancouver: Harry Potter and the Prisoner of Azkaban . Dir.

Alfonso Cuar o n. Based on novel. by J.K. Rowling. Perf. Daniel Radcliffe, Rupert Grint, and saving private ryan Emma Watson. DVD. Has Changed! Warner, 2004. Jane Austen�s Emma . Videocassette. Meridian Broadcasting. New York:

New Video Group, 1996. Kicking Screaming . Dir. Saving Private Length! Jesse Dylan. Writ. Leo Benvenuti and Steve Rudnick. Perf. Will Ferrell and Robert Duvall. DVD. What Is Mock! Universal, 2005. The Sisterhood of the Traveling Pants . Dir. Ken Kwapis.

Based on novel by. Ann Brashares.Perf. Amber Tamblyn, America Ferrera, Blake Lively, and Alexis Bledel. DVD. Warner, Dungaree, 2005. Super Searching the Web . Videocassette.

Lancaster, PA: Classroom Connect, The Wizard of Oz . Dir. Victor Fleming. Based on book by Lyman Frank Baum. Perf. Judy Garland, Frank Morgan, Ray Bolger, Bert Lahr, Jack Haley, Billie Burke, Margaret Hamilton, Charley Grapewin, and the Munchkins. MGM, 1939. VHS.

Warner, 1999. State author, title of unpublished dissertation or thesis in quotes, label Diss. Private Ryan Length! or MA thesis, name of university, and year. Elmendorf, James. �The Military and by wilkie the Mall: Society and Culture in Long Beach, California.� BA. thesis. Private Ryan! Hampshire College, 1995.

Jackson, Marjorie. To Break! �The Oboe: A Study of Its Development and Use.� Diss. Columbia U, 1962.

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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 ryan length, integers in Children's, the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on. An abelian extension of saving, a field is a Galois extension of the field with abelian Galois group.

Class field theory describes the to Break a Man, abelian extensions of a number field in terms of the private ryan length, 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. What Is Mock! v3.00 (February 11, 2008). Corrected; revisions and additions; 163 pages. v3.01 (September 28, 2008). Fixed problem with hyperlinks; 163 pages. Saving Private Ryan Length! v3.02 (April 30, 2009). Fixed many minor errors; changed chapter and page styles; 164 pages. v3.03 (May 29, 2011). Minor fixes; 167 pages. Available at atenolol anxiety, www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph is of the Fork Hut, Huxley Valley, New Zealand.

Copyright c 1996, 1998, 2008, 2009, 2011 J.S. Milne. Saving Private Length! Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Notations. . . Atenolol 50 Mg Anxiety! . . . . . Saving! . . . . . . . . . . Anxiety! . . . . . . . . . . . . Saving! . . . . . . . . . 5 Prerequisites . . . . . . . . . 50 Mg Anxiety! . . Private Ryan Length! . . . . . . Collins! . . . . . . . . . Saving Ryan! . . . . . . . . . . . 5 Acknowledgements . . . Investigation! . . . . . . . . . . . . Private! . Atenolol 50 Mg Anxiety! . . . . . Ryan Length! . . . . Atenolol 50 Mg Anxiety! . . . . . . . . 5 Introduction . . . . . . . . . . . . . . Saving Private Length! . . . . . It Takes To Break A Man Essay! . . . . . . . . . . . . . Saving! . . What! . . . 1 Exercises . Private Length! . . . . . 50 Mg Anxiety! . . . . . . . . . . . . . . . . . . . . . . . . . . Saving Ryan! . . . . . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . . . . . . . . For Dummies! . . . . . . . . . . . . . . . Private! . . . . . . . . . . 7 Ideals in products of rings . . . Anxiety! . . . . . Ryan! . . . . . . . Atenolol 50 Mg Anxiety! . . Private! . . . . . What! . . . . . . Ryan! . The Moonstone By Wilkie! . 8 Noetherian rings . . . . . Private Length! . . . . Atenolol 50 Mg! . . . . . . . . . . Ryan Length! . . Aquinas 5 Ways For Dummies! . . . Ryan! . . . . . . Has Changed Positively! . . . . . Private Ryan! 8 Noetherian modules . . . . . . . . . . . . . . . . . . 50 Mg! . . . . Private Length! . . . . . . . . . . . 10 Local rings . . . Is Mock! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Rings of fractions . . . . . . . . . . . . . . Saving Ryan! . . . How Technology Positively! . . . . . . . . . . . . . . . . . 11 The Chinese remainder theorem . . . . . Saving Length! . . . . . . . . Atenolol 50 Mg! . . . . . . Saving! . . . . . . . 12 Review of tensor products . . . . . . . . . . . . . . . . A Man Essay! . . . . . . . . . . . . Ryan Length! . . 14 Exercise . . . . . . . . . . . . . . . . . . Is Mock Epic! . . . . . . . . . . . Private Ryan Length! . . . Atenolol Anxiety! . . . . . . . 18. 2 Rings of Integers 19 First proof that the saving ryan, integral elements form a ring . . . . . . . . . . Children's Investigation! . . . . . . . Saving Length! . 19 Dedekind�s proof that the integral elements form a ring . . . . . . . . . . . . Education! . . 20 Integral elements . . . . . . . . . Saving Ryan Length! . . . . . . . . Epic! . . . . . . . . . . . . . . . . . 22 Review of private, bases of A-modules . . . . . . . . . . . . . . . . . . . . . . . . . . Investigation! . 25 Review of norms and traces . . . . . . Private Length! . . . To Break Essay! . . . . . . . . . . . . . . . . . . . . 25 Review of bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . Saving Length! . It Takes A Man! . . . Ryan Length! . 26 Discriminants . . It Takes Essay! . . . . . . . . Saving Private Length! . . . . . . . . . . . . . . . Atenolol 50 Mg Anxiety! . . . . . . . . . . . Private Ryan Length! 27 Rings of integers are finitely generated . . . . . . . . . Atenolol 50 Mg Anxiety! . . . . . Length! . . . . . . . . . 29 Finding the Investigation, ring of integers . . . Saving! . . . . . . 5 Ways For Dummies! . . . . . . . . . . . . . . . . . . . . 31 Algorithms for finding the ring of integers . . . . . . . . . . . Saving! . . . . . . . . . . 34 Exercises . . . . . . . . . . For Dummies! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.

3 Dedekind Domains; Factorization 40 Discrete valuation rings . . . . . . Private Ryan! . What Is Mock Epic! . . . . . . Saving Length! . . . . . . . . . . . . . . . . . . 40 Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Unique factorization of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . The Moonstone By Wilkie! . . 43 The ideal class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Discrete valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Integral closures of Dedekind domains . . . . . . . . . . . Saving! . . . . . . . . . The Moonstone Collins! . . . Private Ryan! 51 Modules over Dedekind domains (sketch). . . . . . . . . Atenolol 50 Mg! . . Ryan Length! . . . . The Moonstone By Wilkie! . . . . . . . 52. Factorization in extensions . . . . . . . Saving Private Length! . . . . . . . . . How Technology Positively! . . . . . . . . . . . . . 52 The primes that ramify . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saving Private Ryan! . . . 54 Finding factorizations . . It Takes To Break! . . Saving! . . . . . . . . . . . . . . . For Dummies! . Saving Ryan Length! . . . . . . . . . . . . 56 Examples of factorizations . . . . . . . . . . . . . . . . . . Atenolol! . . . . . . . . . . . 57 Eisenstein extensions . . Private Length! . Is Mock Epic! . . . . . . Length! . . . . . . . . . . . Aquinas 5 Ways For Dummies! . . . . . . . . . . . . 60 Exercises . . . . . . . . Saving Ryan Length! . . . . . . . . . . . . . . Essay! . . . . . . . . . . . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . . . . . . . . . . . . Private Ryan Length! . . . . . . . . . . Education Positively! . . . . . . . . . . Private Length! . . 63 Statement of the main theorem and its consequences . . . . . . . . . . What! . . . . . . 65 Lattices . . . . . . . . Saving Length! . . . . . . . . . . . . . . . . . . . . . . . . . . Aquinas 5 Ways! . . . . . Private! 68 Some calculus . . . . . . . . . . . . . . . . . . . . . . . . Aquinas 5 Ways! . . . . . Saving Ryan Length! . . . . . . . 73 Finiteness of the class number . . . . . . The Moonstone By Wilkie! . . . . . . Saving! . Aquinas 5 Ways! . . . . . . . . . . . . . . Saving Ryan Length! 75 Binary quadratic forms . . . By Wilkie! . . . . . Saving Private Length! . . . . Aquinas For Dummies! . . . . . . . . . . . . . . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ryan Length! . . 78. 5 The Unit Theorem 80 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . Anxiety! . . . . . . . . . 80 Proof that UK is saving length, finitely generated . . . . . . . . . . . . . . . . . . . . . . . . . 82 Computation of the rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 S -units . . . . . . . . . . . . . . . . . . . . Education Investigation! . . . . . . . . . . . . . . . . . . . . Ryan! 85 Example: CM fields . . Atenolol Anxiety! . . . . Saving Private Ryan Length! . . . . . . . . . . . . . . . . . . . . . . . . . . . Collins! 86 Example: real quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Example: cubic fields with negative discriminant . Length! . . . . . . . . . . Collins! . . Saving Ryan! . . . . 87 Finding .K/ . . . . . . . . . . . . . . . . . Aquinas 5 Ways! . . . . . . . . . . . . . . . . . . . 89 Finding a system of fundamental units . . . . . . . . . . . . . . . . . . Saving Private Length! . . The Moonstone Collins! . . . 89 Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Exercises . . . . . . . . . . . . . . . . . Saving Length! . . . . . . . . . . . . . . . . . . . . . Has Changed! 90. 6 Cyclotomic Extensions; Fermat�s Last Theorem. Ryan! 91 The basic results . . . . . . . . . The Moonstone Collins! . Ryan! . . . . . . . What Is Mock! . . . . . . . . . . . . . . . . . Private! . 91 Class numbers of cyclotomic fields . . . . . . . . . . . . . . It Takes A Lot To Break Essay! . . Ryan Length! . . . . . . . . . 97 Units in cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 The first case of Fermat�s last theorem for regular primes . . . What! . . . . . . . . . . 98 Exercises . . . . . . . . . . Saving Private Ryan! . . . . . . . . . . . . Education! . . . . . . . . . . . . . . . . Saving Private! 100. 7 Valuations; Local Fields 101 Valuations . Has Changed Education! . . . . . . . . . . . . . . . Private Ryan! . . . . . . . . . . . . . . . . . . . . . . 101 Nonarchimedean valuations . . . . . . . . 50 Mg! . . . . Length! . . . . . . . . . . . . . . . . . 102 Equivalent valuations . . . . . Education! . . . . . . . . . . . . . . . . . . . Saving! . . . . . . . . 103 Properties of discrete valuations . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Complete list of valuations for the rational numbers . . A Man Essay! . . . . . . . Saving! . . . . . . What Epic! . 105 The primes of a number field . . . . . . . . . . . . . . . . . . . . . . . . . . . Ryan! . 107 The weak approximation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 109 Completions . What! . . . . . . . . . . . Ryan Length! . . . . . . How Technology Has Changed Positively! . . . . . . . . . . . . . . . . . . . 110 Completions in the nonarchimedean case . Saving Ryan Length! . . . . . . . . . . . . Atenolol! . . . . . . . . . 111 Newton�s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Extensions of nonarchimedean valuations . . . . . . . . . . . Saving Private Ryan Length! . . . . . . . . . . 118.

Newton�s polygon . . . . . . . . . . . . . . . . Children's Education! . . . . . . . . . . . . . . Saving Private Length! . . . . 120 Locally compact fields . . . . . . . . . . . . . . . . . . . . . . 5 Ways! . . . . Saving Length! . . . . . 122 Unramified extensions of a local field . . . . . . . . . . . . . . . . . . . . . . . 123 Totally ramified extensions of K . Anxiety! . . . . . . . . . . . . . . . . . . Saving Length! . . . . . . Aquinas For Dummies! . 125 Ramification groups . . . . . . . . . Saving Private! . . . . . . . 50 Mg! . . . . Length! . . . . . Education! . . . . Saving Private Ryan! . . . . 126 Krasner�s lemma and applications . . . . . . . 5 Ways! . . . . . Saving Private Ryan Length! . . . . . Children's Education! . . . . . . . . 127 Exercises . . Saving! . . . . . Children's Education! . . . . . . . . Saving Private Ryan! . . . . The Moonstone By Wilkie Collins! . . . . . . . . . . . Saving Ryan Length! . . . . . . . . 129. 8 Global Fields 131 Extending valuations . . . . . . . . . . . . . . . . . . . . Education! . . . Saving Ryan Length! . . Children's! . . . . . . . 131 The product formula . . . . . . . . . . . . . . Private! . . . . . . . . 5 Ways! . . . . . . . . . . 133 Decomposition groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 The Frobenius element . . Length! . . . . . . . . . . . . . . . . . . . . . . . What Is Mock! . . . . . . 137 Examples . . . . . . . . . . Private! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Computing Galois groups (the hard way) . . . . . . . . . . Aquinas 5 Ways For Dummies! . . Saving Length! . . . . . . . . . . 140 Computing Galois groups (the easy way) . . . . . Children's Education! . . . . . . . . . . . . . . . . . 141 Applications of the Chebotarev density theorem . . . . . . . Saving Private Length! . . Investigation! . . . . . . . . . 146 Exercises . . . . . . . . . . . . . . . . Private Ryan! . . . . . . . . . . . . . . . . The Moonstone Collins! . . . . . . 147. A Solutions to the Exercises 149. B Two-hour examination 155. We use the private, standard (Bourbaki) notations: ND f0;1;2; : : :g; ZD ring of integers; RD field of real numbers; CD field of complex numbers; Fp D Z=pZD field with p elements, p a prime number.

For integers m and n, mjn means that m divides n, i.e., n 2mZ. Throughout the notes, p is a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . The empty set is denoted by ;. The cardinality of 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 positively 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 saving private ryan length, the end and in footnotes, I shall refer to the following of a Lot a Man, my course notes (available at www.jmilne.org/math/): FT Fields and Galois Theory, v4.22, 2011.

GT Group Theory, v3.11, 2011. CFT Class Field Theory, v4.01, 2011. I thank the following for providing corrections and comments for earlier versions of these notes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Francesc Castella?; Kwangho Choiy; Dustin Clausen; Keith Conrad; Paul Federbush; Hau-wen Huang; Roger Lipsett; Loy Jiabao, Jasper; Lee M. Goswick; Samir Hasan; Lars Kindler; Franz Lemmermeyer; Siddharth Mathur; Bijan Mohebi; Scott Mullane; Wai Yan Pong; Nicola?s Sirolli; Thomas Stoll; Vishne Uzi; and others. PARI is an open source computer algebra system freely available from http://pari.math.u- bordeaux.fr/. FERMAT (1601�1665). Stated his last �theorem�, and proved it for mD 4. He also posed the problem of finding integer solutions to the equation, X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the units in Z? p A?. The English mathemati- cians found an algorithm for solving the problem, but neglected to prove that the algorithm always works.

EULER (1707�1783). He introduced analysis into the study of the prime numbers, and he discovered an early version of the quadratic reciprocity law. LAGRANGE (1736�1813). He found the saving private length, complete form of the quadratic reciprocity law: D .?1/.p?1/.q?1/=4; p;q odd primes, and he proved that the algorithm for solving (1) always leads to a solution, LEGENDRE (1752�1833). He introduced the �Legendre symbol� m p. , and gave an incom- plete proof of the quadratic reciprocity law. He proved the following local-global principle for quadratic forms in three variables over Q: a quadratic form Q.X;Y;Z/ has a nontrivial zero in Q if and only if it has one in R and the congruence Q 0 mod pn has a nontrivial solution for all p and n. GAUSS (1777�1855). He found the first complete proofs of the the moonstone by wilkie collins, quadratic reciprocity law. He studied the Gaussian integers Z?i ? in order to find a quartic reciprocity law. Ryan! He studied the classification of binary quadratic forms over Z, which is closely related to the problem of aquinas 5 ways, finding the class numbers of quadratic fields.

DIRICHLET (1805�1859). He introduced L-series, and used them to prove an analytic for- mula for the class number and a density theorem for private the primes in an arithmetic progression. He proved the following �unit theorem�: let ? be a root of a monic irreducible polynomial f .X/ with integer coefficients; suppose that f .X/ has r real roots and 2s complex roots; then Z??? is a finitely generated group of rank rC s?1. KUMMER (1810�1893). He made a deep study of the arithmetic of Children's, cyclotomic fields, mo- tivated by a search for higher reciprocity laws, and showed that unique factorization could be recovered by the introduction of �ideal numbers�. He proved that Fermat�s last theorem holds for regular primes. HERMITE (1822�1901). He made important contributions to quadratic forms, and he showed that the roots of a polynomial of degree 5 can be expressed in terms of elliptic functions.

EISENSTEIN (1823�1852). He published the first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (1823�1891). He developed an alternative to Dedekind�s ideals. He also had one of the most beautiful ideas in mathematics for saving private generating abelian extensions of number fields (the Kronecker liebster Jugendtraum). RIEMANN (1826�1866). Studied the Riemann zeta function, and made the Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion. DEDEKIND (1831�1916). He laid the modern foundations of algebraic number theory by finding the correct definition of the ring of integers in a number field, by proving that ideals factor uniquely into products of aquinas 5 ways for dummies, prime ideals in such rings, and by showing that, modulo principal ideals, they fall into finitely many classes.

Defined the zeta function of a number field. WEBER (1842�1913). Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (1861�1941). He gave the first definition of the field of p-adic numbers (as the set of infinite sums. n, an 2 f0;1; : : : ;p?1g). HILBERT (1862�1943).

He wrote a very influential book on algebraic number theory in 1897, which gave the first systematic account of the theory. Some of his famous problems were on number theory, and have also been influential. TAKAGI (1875�1960). He proved the fundamental theorems of abelian class field theory, as conjectured by Weber and Hilbert. NOETHER (1882�1935). Together with Artin, she laid the foundations of modern algebra in which axioms and private conceptual arguments are emphasized, and she contributed to the classification of atenolol 50 mg anxiety, central simple algebras over number fields.

HECKE (1887�1947). Introduced HeckeL-series generalizing both Dirichlet�sL-series and Dedekind�s zeta functions. ARTIN (1898�1962). He found the �Artin reciprocity law�, which is the main theorem of class field theory (improvement of ryan length, Takagi�s results). Introduced the Artin L-series. HASSE (1898�1979). He gave the first proof of Essay, local class field theory, proved the Hasse (local-global) principle for all quadratic forms over number fields, and contributed to saving private ryan length the classification of central simple algebras over number fields. BRAUER (1901�1977). Defined the Brauer group, and contributed to the classification of central simple algebras over number fields. WEIL (1906�1998).

Defined the Weil group, which enabled him to give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (1909�84). The main statements of class field theory are purely algebraic, but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. What Epic! With his introduction of ide?les he was able to ryan give a natural formulation of class field theory for infinite abelian extensions. IWASAWA (1917�1998). He introduced an important new approach into algebraic number theory which was suggested by the theory of curves over finite fields.

TATE (1925� ). He proved new results in the moonstone by wilkie, group cohomology, which allowed him to give an elegant reformulation of class field theory. With Lubin he found an explicit way of generating abelian extensions of local fields. LANGLANDS (1936� ). The Langlands program2 is a vast series of conjectures that, among other things, contains a nonabelian class field theory. 2Not to be confused with its geometric analogue, sometimes referred to saving ryan as the geometric Langlands pro- gram, which appears to lack arithmetic significance. Introduction It is greatly to be lamented that this virtue of the [rational integers], to be decomposable into prime factors, always the same ones for a given number, does not also belong to the [integers of education, 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 saving private length, essentially unique. Consider more generally an integral domain A. An element a 2A is said to be a unit if. it has an inverse in A (element b such that ab D 1D ba). I write A for the multiplicative group of units in A. An element of A is how technology has changed education, said to saving private ryan prime if it is neither zero nor a unit, and if. If A is a principal ideal domain, then every nonzero element a of A can be written in the form, aD u1 n; u a unit; i a prime element; and this factorization is unique up to order and replacing each i with an associate, i.e., with its product with a unit. Children's Education! Our first task will be to discover to what extent unique factorization holds, or fails to hold, in number fields.

Three problems present themselves. Saving Ryan! First, factorization in a field only makes sense with respect to a subring, and so we must define the �ring of integers� OK in our number field K. Secondly, since unique factorization will fail in general, we shall need to find a way of measuring by how much it fails. Finally, since factorization is only considered up to units, in order to fully understand the arithmetic of K, we need to understand the structure of the group of units UK in OK . THE RING OF INTEGERS. Let K be an algebraic number field. Each element ? of K satisfies an equation. ?nCa1? n?1 C Ca0 D 0. with coefficients a1; : : : ;an in Q, and ? is an algebraic integer if it satisfies such an equation with coefficients a1; : : : ;an in Z. We shall see that the algebraic integers form a subring OK of K. The criterion as stated is difficult to apply. We shall show (2.11) that ? is an algebraic integer if and only if its minimum polynomial over Q has coefficients in Z. Consider for example the field K D Q? p d?, where d is a square-free integer. The. minimum polynomial of It Takes Essay, ? D aCb p d , b � 0, a;b 2Q, is. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and so ? is an algebraic integer if and only if. 2a 2 Z; a2?b2d 2 Z: From this it follows easily that, when d 2;3 mod 4, ? is an algebraic integer if and only if a and b are integers, i.e., and, when d 1 mod 4, ? is an algebraic integer if and only if a and b are either both integers or both half-integers, i.e., For example, the minimum polynomial of 1=2C p 5=2 is X2?X ?1, and so 1=2C. is an algebraic integer in Q? p 5?.

Let d be a primitive d th root of 1, for 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. Private Ryan! A nonzero element of an integral domain A is said to be irreducible if it is not a unit, and can�t be written as a product of two nonunits. Collins! For example, a prime element is (obviously) irreducible. A ring A is a unique factorization domain if every nonzero element of A can be expressed as a product of irreducible elements in essentially one way. Is the ring of integers OK a unique factorization domain? No, not in general! We shall see that each element of OK can be written as a product of irreducible elements (this is true for all Noetherian rings), and so it is the uniqueness that fails. For example, in Z? p ?5? we have.

6D 2 3D .1C p ?5/.1?. To see that 2, 3, 1C p ?5, 1?. p ?5 are irreducible, and no two are associates, we use the. p ?5 7! a2C5b2: This is multiplicative, and it is easy to see that, for ? 2OK , Nm.?/D 1 � ? N? D 1 � ? is a unit. Saving Ryan Length! (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. p ?5/D 6. Thus Nm.?/D 1;2;3, or 6. In the. first case, ? is a unit, the second and third cases don�t occur, and in the fourth case ? is 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. 50 Mg! has no solution with a;b 2 Z. Saving Private Length! 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 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 atenolol, 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. Private Ryan Length! Thus an ideal factor a is a set of elements of OK such that. 0 2 aI a;b 2 a) a?b 2 aI a 2 a) ab 2 a for all b 2OK I. it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of 50 mg, you will recognize that an ideal factor is what we now call an ideal, and a prime ideal factor is a prime ideal. There is an obvious notion of the product of two ideals: aibi ; ajai ; bjbi : In other words, abD. nX aibi j ai 2 a; bi 2 b. One see easily that this is again an ideal, and that if. aD .a1; . ;am/ and bD .b1; . ;bn/ then a bD .a1b1; . ;aibj ; . ;ambn/: With these definitions, one recovers unique factorization: if a � 0, then there is an essentially unique factorization: .a/D p1 pn with each pi a prime ideal. In the above example, .6/D .2;1C p ?5/.2;1?.

In fact, I claim. .2;1C p ?5/.2;1?. .3;1C p ?5/.3;1?. .2;1? p ?5/.3;1?. For example, .2;1C p ?5/.2;1?. p ?5;6/. Since every gen- erator is divisible by 2, we see that. .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?. p ?5/ are all prime. Length! 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 Children's 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 ryan, 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 epic, 1 in K form a finite group .K/, and that. OK .K/Z r (as an abelian group); moreover, we shall find r: One motivation for the development of algebraic number theory was the attempt to prove Fermat�s last �theorem�, i.e., when m 3, there are no integer solutions .x;y;z/ to the equation. with all of x;y;z nonzero. WhenmD 3, this can proved by the method of �infinite descent�, i.e., from one solution, you show that you can construct a smaller solution, which leads to a contradiction3. The proof makes use of the factorization. Y 3 DZ3?X3 D .Z?X/.Z2CXZCX2/; and it was recognized that a stumbling block to proving the theorem for larger m is that no such factorization exists into ryan polynomials with integer coefficients of degree 2. This led people to look at more general factorizations. In a famous incident, the French mathematician Lame? gave a talk at the Paris Academy in 1847 in which he claimed to prove Fermat�s last theorem using the what is mock, 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 ryan length, Lame?�s proof in which he assumed that, in order to show that each factor .z ? iy/ is a pth power, it suffices to show that the factors are relatively prime in pairs and their product is a pth power. In fact, Lame? couldn�t justify his step (Z?? is not always a principal ideal domain), and Fermat�s last theorem was not proved for almost 150 years. For Dummies! However, shortly after Lame?�s embarrassing lecture, Kummer used his results on the arithmetic of the fields Q?? to prove Fermat�s last theorem for all regular primes, i.e., for all primes p such that p does not divide the class number of Q?p?. Another application is to finding Galois groups. The splitting field of a polynomial f .X/ 2Q?X? is a Galois extension of Q. In a basic Galois theory course, we learn how to 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 saving private ryan length, second develops the whole subject algorithmically. Cohen�s book is the more useful as a supplement to this course, but wasn�t available when these notes were first written. While the books are concerned with more-or-less practical algorithms for fields of small degree and small discriminant, Lenstra (1992) concentrates on finding �good� general algorithms. 3The simplest proof by 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 algebraic number theory, con- centrating on the efforts to what epic prove Fermat�s last theorem. The notes in Narkiewicz 1990 document the origins of most significant results in algebraic number theory.

Lemmermeyer 2009, which explains the origins of �ideal numbers�, and other writings by the same author, e.g., Lemmermeyer 2000, 2007. Private! 0-1 Let d be a square-free integer. How Technology Education! Complete the verification that the ring of integers in Q? p d? is as described. 0-2 Complete the verification that, in Z? p ?5?, .6/D .2;1C p ?5/.2;1?. is a factorization of .6/ into a product of prime ideals. CHAPTER 1 Preliminaries from Commutative. Length! 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 collins 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 the ?i . It consists of all polynomials in the ?i with coefficients in A, i.e., elements of the form X.

ai1. im? i1 1 . ? im m ; ai1. im 2 A: We also refer to A??1; . ;?m? as the A-subalgebra of B generated by the ?i , and when B D A??1; . ;?m? we say that the ?i generate B as an A-algebra. For elements a1;a2; : : : of A, we let .a1;a2; : : :/ denote the smallest ideal containing the ai . It consists of finite sums. P ciai , ci 2 A, and private length 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 the moonstone collins, fact, it is the smallest ideal containing both a and b. If aD .a1; . ;am/ and bD .b1; . ;bn/, then aCbD .a1; . ;am;b1; . ;bn/: Given an ideal a in A, we can form the quotient ring A=a. Let f WA! A=a be the homomorphism a 7! aCa; then b 7! f ?1.b/ defines a one-to-one correspondence between the ideals of A=a and the ideals of A containing a, and. 1See also the notes A Primer of Commutative Algebra available on my website. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. A proper ideal a of A is prime if ab 2 a) a or b 2 a. An ideal a is prime if and only if the quotient ring A=a is an integral domain. A nonzero element of A is said to be prime if ./ is a prime ideal; equivalently, if jab) ja or jb.

An ideal m in A is maximal if it is maximal among the proper ideals of A, i.e., if m�A and there does not exist an ideal a � A containing m but distinct from it. An ideal a is maximal if and only if A=a is a field. Every proper ideal a of A is contained in a maximal ideal � if A is Noetherian (see below) this is obvious; otherwise the proof requires Zorn�s lemma. In particular, every nonunit in A is contained in a maximal ideal. There are the implications: A is ryan length, a Euclidean domain) A is a principal ideal domain ) A is a unique factorization domain (see any good graduate algebra course). Ideals in products of rings.

PROPOSITION 1.1 Consider a product of rings AB . 50 Mg! If a and b are ideals in A and B respectively, then ab is an saving private length 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. 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. Atenolol 50 Mg Anxiety! 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 saving ryan, 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 aquinas for dummies, a j such that aj is a prime ideal in Aj and ai DAi for i � j: A ring A is Noetherian if every ideal in A is finitely generated.

PROPOSITION 1.3 The following conditions on a ring A are equivalent: (a) A is Noetherian. (b) Every ascending chain of ideals. eventually becomes constant, i.e., for length 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 . How Technology Education! PROOF. (a) (b): Let a D S. ai ; it is an ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and so an D anC1 D D a. (b) (c): Let a1 2 S . If a1 is not a maximal element of S , then there exists an a2 2 S such that a1 a2. If a2 is not maximal, then there exists an a3 etc.. From (b) we know that this process will lead to a maximal element after only finitely many steps. (c) (a): Let a be an ideal in A, and let S be the set of finitely generated ideals contained in saving private ryan length, a. Then S is nonempty because it contains the zero ideal, and so it contains a maximal element, say, a0 D .a1; : : : ;ar/. If a0 � a, then there exists an element a 2 ar a0, and .a1; : : : ;ar ;a/ will be a finitely generated ideal in a properly containing a0. This contradicts the definition of a0. 2. A famous theorem of Hilbert states that k?X1; . ;Xn? is Noetherian. In practice, al- most all the rings that arise naturally in algebraic number theory or algebraic geometry are Noetherian, but not all rings are Noetherian. For example, the ring k?X1; : : : ;Xn; : : :? of polynomials in an infinite sequence of symbols is not Noetherian because the chain of ideals. never becomes constant.

PROPOSITION 1.4 Every nonzero nonunit element of a Noetherian integral domain can be written as a product of irreducible elements. PROOF. We shall need to It Takes a Lot to Break a Man use that, for elements a and b of an integral domain A, .a/ .b/ � bja, with equality if and only if b D aunit: The first assertion is obvious. For the second, note that if a D bc and b D ad then a D bc D adc, and so dc D 1. Hence both c and d are units. Suppose the statement of the proposition is saving length, false for a Noetherian integral domain A. Then there exists an Children's element a 2 A which contradicts the statement and is such that .a/ is maximal among the ideals generated by such elements (here we use that A is Noetherian). Since a can not be written as a product of irreducible elements, it is not itself irreducible, and so a D bc with b and c nonunits.

Clearly .b/ .a/, and the ideals can�t be equal for otherwise c would be a unit. From the maximality of .a/, we deduce that b can be written as a product of saving ryan, irreducible elements, and similarly for c. Thus a is a product of a Lot Essay, 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 ?. Thus O is not Noetherian. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Let A be a ring. An A-module M is said to be Noetherian if every submodule is saving private ryan, finitely generated. PROPOSITION 1.6 The following conditions on an A-module M are equivalent: (a) M is aquinas 5 ways, 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 saving length, Proposition 1.3. 2. PROPOSITION 1.7 Let M be an what is mock A-module, and let N be a submodule of M . If N and M=N are both Noetherian, then so also is M . PROOF. I claim that if M 0 M 00 are submodules of M such that M 0N DM 00N and M 0 and M 00 have the same image in M=N , then M 0 DM 00. To see this, let x 2M 00; the second condition implies that there exists a y 2M 0 with the same image as x inM=N , i.e., such that x?y 2N . Saving Private Length! 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 Investigation, Noetherian, the image of the chain in M=N becomes constant, and private length if N is Noetherian, the intersection of the chain with N becomes constant. Now the claim shows that the atenolol anxiety, 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 private length, generated by a single element, then M A=a for some ideal a in A, and the statement is obvious. We argue by induction on the minimum number n of generators ofM . SinceM contains a submoduleN generated by n?1 elements such that the It Takes a Lot to Break, quotient M=N is saving private ryan length, generated by a single element, the statement follows from (1.7). 2. A ring A is said to local if it has exactly one maximal ideal m. In this case, A D Arm (complement of m in A). LEMMA 1.9 (NAKAYAMA�S LEMMA) Let A be a local Noetherian ring, and let a be a proper ideal in Children's, 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 saving private, a submodule of M such that N CaM DM , then N DM: Rings of how technology education, 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. Ryan Length! 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. To Break! Con- sider mCN , m 2M . From the assumption, we can write. aimi , with ai 2 a, mi 2M: and so mCN 2 a.M=N/: 2. The hypothesis that M be finitely generated in the lemma is essential. For example, if A is a local integral domain with maximal ideal m � 0, then mM DM for any field M containing A but M � 0. Rings of saving length, 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 It Takes to Break Essay, k?X?: Let A be an integral domain with field of fractions K. A subset S of A is said to be multiplicative if 0 � S , 1 2 S , and S is closed under multiplication. If S is a multiplicative subset, then we define.

S?1AD fa=b 2K j b 2 Sg: It is obviously a subring of K: EXAMPLE 1.10 (a) Let t be a nonzero element of A; then. St def D f1,t ,t2. g. is a multiplicative subset of A, and we (sometimes) write At for S?1t A. For example, if d is a nonzero integer, then2 Zd consists of those elements of saving private length, 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 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. By Wilkie Collins! Let a be an ideal in S?1A. Clearly .aA/e a because aA a and a is an ideal in saving ryan, S?1A. For the reverse inclusion, let b 2 a. We can write it b D a=s with a 2 A, s 2 S . Then aD s .a=s/ 2 aA, and so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . Clearly .S?1p/A p. A Man! 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. Saving Private Length! 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 aquinas 5 ways for dummies, A disjoint from S; and (1.11) shows that the two maps are inverse. 2. EXAMPLE 1.13 (a) If p is a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). (b) We list the private ryan length, 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 how technology has changed positively, theorem: let d1; . ;dn be integers, relatively prime in pairs; then for any integers x1; . ;xn, the saving ryan, congruences. The Chinese remainder theorem. have a simultaneous solution x 2 Z; moreover, if x is one solution, then the other solutions are the integers of the form xCmd with m 2 Z and d D. We want to translate this in terms of ideals. Integersm and n are relatively prime if and only if .m;n/D Z, i.e., if and only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to be relatively prime if aCbD A. If m1; . ;mk are integers, then T .mi / D .m/ where m is the the moonstone collins, 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 /. Private Ryan! 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 atenolol 50 mg, 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 i2.ai Cbi /D 1, and lies in a1C. Q i2 ai , and so.

We can now apply the theorem in the case nD 2 to obtain an element y1 of A such that. y1 1 mod a1; y1 0 mod Y. These conditions imply. y1 1 mod a1; y1 0 mod aj , all j 1: Similarly, there exist elements y2; . ;yn such that. yi 1 mod ai ; yi 0 mod aj for saving ryan j � i: The element x D P xiyi now satisfies the requirements. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. It remains to prove that T. ai . We have already noted that T. ai . First suppose that nD 2, and let a1Ca2 D 1, as before. For c 2 a1a2, we have. c D a1cCa2c 2 a1 a2. which proves that a1 a2 D a1a2. We complete the proof by induction. This allows us to assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . The theorem extends to A-modules. THEOREM 1.15 Let a1; . ;an be ideals in A, relatively prime in pairs, and let M be an A-module.

There is an exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to use tensor products, which I now review. Review of tensor products. Let M , N , and collins P be A-modules. A mapping f WM N ! P is said to be A-bilinear if. f .mCm0;n/D f .m;n/Cf .m0;n/ f .m;nCn0/D f .m;n/Cf .m;n0/ f .am;n/D af .m;n/D f .m;an/ 9=; all a 2 A; m;m0 2M; n;n0 2N: i.e., if it is linear in 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. Private Ryan Length! The tensor product exists, and is unique (up to a unique isomorphism making the obvious diagram commute).

We denote it by Education, M ?AN , and we write .m;n/ 7! m?n for f . The pair .M ?AN;.m;n/ 7!m?n/ is characterized by each of the following two conditions: (a) The mapM N !M ?AN is A-bilinear, and private length any other A-bilinear mapM N ! P is of the is mock, form .m;n/ 7! ?.m?n/ for a unique A-linear map ?WM ?AN ! P ; thus. Private Length! 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 what a field k of dimensions m and n respectively, then V ?kW is saving private length, a vector space over k of dimension mn. Let ?WM !M 0 and atenolol ?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 private, the elements m0?n0, m0 2 M 0, n0 2 N 0. By assumption m0 D ?.m/ for some m 2M and n0 D ?.n/ for some n 2 N , and som0?n0 D ?.m/??.n/D .???/.m?n/. Therefore the image of ??? contains a set of generators for M 0?AN 0 and so it is equal to it. 50 Mg! 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 private, 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 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 Education Investigation 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 private ryan 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 by wilkie collins 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 ryan length, a field k and ? is an atenolol arbitrary field extension of k. According to the primitive element theorem (FT 5.1), K D k??? for some ? 2K. Let f .X/ be the minimum polynomial of ?. By definition this means that the map g.X/ 7! g.?/ determines an isomorphism.

Hence K?k? ' .k?X?=.f .X///?k? '??X?=.f .X// by (3) and (4). Because K is separable over k, f .X/ has distinct roots. Therefore f .X/ factors in ??X? into monic irreducible polynomials. that are relatively prime in pairs. We can apply the Chinese Remainder Theorem to deduce that. Finally, ??X?=.fi .X// is a finite separable field extension of ? of degree degfi . Thus we have proved the following result: THEOREM 1.18 Let K be a finite separable field extension of k, and let ? be an arbitrary field extension. Then K?k? is a product of finite separable field extensions of ?, If ? is a primitive element for K=k, then the image ?i of ? in ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the saving private ryan, minimum polynomials for ? and ?i respectively, then. EXAMPLE 1.19 Let K DQ??? with ? algebraic over Q. Then. C?QK ' C?Q .Q?X?=.f .X///' C?X?=..f .X//' Yr. iD1 C?X?=.X ??i / Cr : Here ?1; : : : ;?r are the conjugates of ? in C. The composite of ? 7! 1??WK!

C?QK with projection onto It Takes a Lot Essay the i th factor is. We note that it is essential to assume in (1.18) that K is separable over k. If not, there will be an ? 2K such that ?p 2 k but ? � k, and the ring K?kK will contain an element ? D .??1?1??/� 0 such that. ?p D ?p?1?1??p D ?p.1?1/??p.1?1/D 0: Hence K?kK contains a nonzero nilpotent element, and so it can�t be a product of fields. NOTES Ideals were introduced and saving private studied by Education, 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.

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Free Essays on Functionalism And Marxism. scientific theories based on critical awareness of society. The two main stems of sociological concepts are Positivism and ryan, Phenomenology. Both Functionalism and collins, Marxism are Positivist theories. This means that philosophies are built by ryan length using scientific research methods to create structural perspectives. Positivism.

Functionalism and Marxism are traced back to theories adopted by sociologists in the nineteenth century. Marxism came from the German philosopher Karl Marx (1818-1883), whereas Functionalism was originally derived by Auguste Compte (1798-1857). It was then developed further by Emile Durkheim (1858-1917). ?P1 Explain the principal sociological principles Functionalism : Functionalism (or structural functionalism ) is the perspective in Children's Investigation, sociology according to which society. Essay Title: Compare and Contrast Functionalism and Marxism. Functionalism and length, Marxism are both known to be structural perspectives, due to the fact that they concentrate on a group of people rather than on the individual himself. How Technology Positively? Although very similar the two are different in very distinct ways, in private ryan, fact Functionalism falls under the Children's Investigation sub-heading of length, consensus structuralism. Structural functionalism , or simply functionalism , is a framework for what is mock epic building theory that sees society as a complex system whose parts work together to promote solidarity and stability.[1] This approach looks at society through a macro-level orientation, which is a broad focus on the social structures. ?�Assess the usefulness of consensus theories such as functionalism , to our understanding of contemporary society�.

Functionalism is a structural consensus theory; it explains society in saving private ryan length, its totality, and assumes that the social world exists in how technology education positively, a state of harmony. For functionalists such as Durkheim. explanations as to how we can best understand human society. Saving Private Length? The main theoretical perspectives covered in epic, GCSE Sociology are; Functionalism Marxism Feminism The New Right Functionalism Functionalists believe that society can best be compared to a living organ, in which institutions and people all have. Maria Angela P. Taruc A � 51 1. Saving Private? One of the weaknesses of the aquinas 5 ways idea of saving private length, Marxism is its inconveniency of defining the needs of its people. What Is Mock Epic? And since it promoting equality, meaning no more monetary rewards, people will not be encouraged to work more, spend less hours on the job and ryan, there would be no. Marxism Philosophy and Modern Communism. Marx/ Marxism Karl Marx is known as one of the education positively great intellects of our time. Ryan Length? His philosophies rejected the concept of Western democracy because he alleged that they enforced selfishness and competitiveness among all. Over one third of the world�s population follows Marx�s theories, known mostly.

problems, which Comte and after him, Spencer and Taine, had discussed and mapped, became a precise and concrete study only when the attack of militant Marxism made its conclusions a burning issue, and so made the search for evidence more zealous and the attention to Education, method more intense. � Isaiah Berlin. ?� Marxism is appropriate for the analysis of the Caribbean.� Critically evaluate this statement. (25 marks) Marxism is a sociological Macro theory, alternative to functionalism , developed by Karl Marx. Marxism is private, a conflict theory based on Children's, the relationship between the bourgeoisie and the proletariat. range theories, applicable to saving length, limited ranges of data transcend sheer description of social phenomena. Merton argued that the central orientation of functionalism is in interpreting data by their consequences for larger structures, in which they are implicated. To Break A Man? The American Marx scholar Hal Draper once remarked. Critically Evaluate the Contributions of Functionalism to the Study of Society. Critically evaluate the contributions of functionalism to the study of saving private, society. Children's Education? Functionalist theory is one of the major theoretical perspectives in sociology. It can be argued that the functionalist theory has made a significant contribution to the study of society.

It originates from the work of Emile. As a sociological discipline, functionalism is private ryan length, counterposed to has changed, Marxism . However it shares with Marxism the importance of �totality� and the corresponding view that scientific inquiry is based upon the interdependence of parts within a whole. It is important to saving ryan, distinguish why the Marxian use of the totality. perspective suggests that society cannot be analyzed from how technology has changed a top-down view, but must be viewed from the saving private individual level. THE MACRO THEORIES FUNCTIONALISM - The perspective of examining society from the wider view of society itself. This theory is concerned with social order and its maintenance. . shown to be the most durable as well as flexible than Mark believed.

Many other types of governments such as Neo- Marxism has received criticism as being a variety of 5 ways, left wing functionalism , replacing a society that exists for the benefit of all, Jock Young states that this statement �for the benefit. analyse it, understand it and evaluate it. With theories and theorist on society available I will focus on two of the theories which are Marxism by saving ryan Karl Marx amp; Functionalism by Skinner. Karl Marx, the man was a genius. The man was not blind, most people are blind, we look but we cannot see and the moonstone by wilkie collins, when we do.

Evaluate the usefulness of saving private ryan, Functionalist theories to our understanding of crime and deviance (40 marks) other major theoretical perspectives such as Marxism , no specific Functionalist criminology exists to speak of, with its own individual interpretations of criminal statistics, the source of criminality and potential policy solutions. Rather functionalism takes a passing look at what is mock epic the issues of deviance. Structuralism vs. Private Ryan Length? Functionalism 1 Structuralism vs. Functionalism Andrew Beasley National University Structuralism vs. Functionalism 2 Structuralism vs. Functionalism Is it better to study the mind through life experiences or through the way an organism adapts to its environment. Structuralism vs. Functionalism When Wilhelm Wundt�s work helped psychology finally break away and became a separate science, the how technology has changed positively argument over how to private ryan length, describe and explain the human mind and behavior erupted. Structuralism surfaced followed closely by other theories ready to take over psychology. . those social systems. What? Sociologists talk about theories but many of them see them from different perspectives.

Structural perspective like Functionalism and Marxism ; examine the way in which society act as a whole. Structural perspectives tend to see human activity as a product of the social structure. Conflict Theory and Functionalism - Essat. Conflict Theory and saving private length, Functionalism There are three main theories of sociology; functionalism , conflict theory and symbolic interactionism. This paper will focus on two of for dummies, those theories, functionalism and conflict theory. The objective is to delineate the assumptions of two out of the saving private three theoretical. The Main Ideas of Marxism, Functionalism and Internationalism. Outline the main ideas of Marxism , Functionalism and Internationalism (social action theory). Are these ideas complementary or contradictory? Sociology is the academic study of individuals and groups in the society. Sociology is all about the ways in which people form the society which they live in.

Marxism and the Role of Education. values and morals of society. Yet there are three sociological theories that differ greatly between them on the role of education. These are Functionalism , Marxism and Liberalism. Examine the Marxist view that the role of the education system is to Investigation, reproduce and justify the saving ryan length existing class structure. In. Introduction to Sociological Perspectives. Introduction The sociological perspective is defined by three philosophical traditions (or paradigms): structure- functionalism , Marxism , and symbolic interactionism.

Structure- functionalism focuses on how society is organized and the moonstone, how social institutions meet the private ryan length needs of people living within a collectivity. the spiritual world and giving meaning to the divine. On the other hand latent functions of religion are unintended and hidden. The believers of functionalism suggest that religion is a requirement for society and is mock, individual because it serves both manifest and latent functions. The manifest functions. Functionalism and Marxism: Sociological Perspectives. Sociological Concepts and Perspectives: Functionalism and saving private, Marxism In this essay I am going to compare and contrast Functionalism and aquinas 5 ways, Marxism . They are both sociological perspectives which have theories about society and saving ryan length, the people that live within it. They attempt to explain how society influences people. potential -most human behaviour is learned, therefore, society can have a great influence Schools of Thought in Sociology Structural Functionalism -the idea that human societies have basic needs that must be met -the following are the fundamental requirements to ensure that society functions. Assess the contribution of Marxism to our understanding of Children's, families and households.

?Assess the contribution of Marxism to saving private ryan, our understanding of how technology positively, families and households (24 marks) Marxists see all society�s institutions as helping to maintain class inequality and Capitalism. Therefore, the main contribution of Marxism to families and households has been to explain how the family. ? Functionalism and crime: In this essay I will be talking about the functionalist perspective on crime and deviance and be comparing it with the Marxist view. Private Ryan? The main functionalist theories I will be examining are Merton�s strain theory, Cohen�s status frustration and Children's Education, Cloward and Ohlin�s three subcultures. A Critical outline of the saving ryan length main features of It Takes Essay, Functionalism, Symbolic Interactionism And Marxism. are concerned with how the different parts of society contribute towards the private length whole. All members of society are imposed with common values and norms. Marxism sees the overall structure of society primarily determined or influenced by the economic system, the means of production, such as the land, factories.

Evaluate and Analyse the Relationship Between Religion and Social Change. (40 Marks) hard work is necessary to gain gods favour is a central belief in most religious doctrines. Religion for social change; Neo- Marxism shares many ideas with traditional Marxism however there is disagreement over the role of is mock, religion in society. Gramsci was a particularly influential member of the neo. �Every Sociological Perspective Has Its Limitations; However Some Are More Useful Than Others to Our Understanding of Society�. Assess This View. are the structional view and the social action view. From these �macro� and �micro� perspectives stem a range of sociological views such as Functionalism , Marxism , Feminisist theories and social action theories, that all take upon different views and ideas of society. It is through these perspectives. Understanding Functionalism, Marxism and saving private length, Liberalism. values and morals of a Man Essay, society.

Yet there are three sociological theories that differ greatly between them on the role of education. These are Functionalism , Marxism and Liberalism. Functionalists view the saving private ryan role of education as a means of socialising individuals and to integrate society, to keep society. the 18th Century. 50 Mg? The three founding fathers of sociology are Max Weber, Karl Marx and Emile Durkheim. Karl Marx created a key theory of length, sociology; Marxism . Marxists focus on society as being based on what is mock epic, class conflict and divided into two groups, the proletariats and the bourgeoisies who are set out to exploit. P1- Explain the Principle Sociological Perspectives. of society. In this assignment I will be explaining the different sociological perspectives which provide different models of society. Functionalism Functionalism looks at social structures and the role they have in society. They believe that each social structure is essential for interests of saving private, society.

One of the It Takes a Lot Essay main assumptions of saving private, Marxism : Contradiction and Conflict. The theory of Marxism is a fundamental alternative to functionalism . What Epic? It was largely used and appreciated during the 1970s, due to the decline of functionalism and ryan, the assurance that it could offer answers which functionalism could not provide. Also, Marxism was more in sync with that era. It takes its. Explaining functionalism, marxism, and symbolic interactionism and some differences between these 3 sociological perspectives. Functionalism focuses on what is good for the whole of society. Functionalists took a similar way as biologists to explain this perspective.

Social systems were dissected into their parts, or institutions (family, education, economy, polity, and religion), and these parts were examined to find out. Assess the View That Marxism Is Not Relevant Today. Assess the view that Marxism is not relevant today Marxism is a structural theory, seeing society and all its institutions controlled by anxiety the bourgeoisie to serve the ruling class interests. Traditional Marxism sees society divided into two classes: the ruling capitalist (bourgeoisie) class who own. Evaluate the Postmodernist Explanation of the Role and Function of Religion in Contemporary Society. (40 Marks) of the role and function of religion has been criticised to some extent.

Firstly postmodernism criticises other sociological explanations, functionalism and Marxism , referring to them as �meta-narratives�, but postmodernism is a meta-narrative itself since it is a theory, interpreting the role of religion. An Examination of the Communist Revolution of China as a Representation of Marxism and Maoism. many, Communism and Marxism are interchangeable, despite the differences between the two. Communal societies have existed long before the Industrial Revolution, while Marxism was only created during the saving mid-nineteenth century after the publication of The Communist Manifesto. Marxism goes beyond just the. strenghts and limitations of It Takes a Lot a Man, functionalism. Success---- Functionalism is seen as a macro-scale approach to society; it sees society as a whole rather than looking at parts of private, it. Functionalism also uses biological analogy to explain society, this means all the institutions work together to make society. How Technology Has Changed? This is particularly useful when observing. sociological accounts of ethnicity have been classified and critically differentiated by Sinisa Malesevic (2004).

Classical sociology, neo- Marxism , functionalism , symbolic interactionism, sociobiology, rational-choice theory, elite theory, neo-Weberian approaches, and antifoundationalist positions have. �Marxism Is No Longer Relevant to Our Understanding of Crime Deviance in Society� etc. Marx developed the idea of Marxism (a conflict theory between Upper and Lower social classes) in the 18th Century, when social classes were very clearly defined- the �Bourgeois� and the �Proletariat�. For this very reason, what is known as �Traditional Marxism � is saving length, now quite evidently outdated. Functionalism Batman is the hero Gotham deserves, but not the one it needs right now. So we'll hunt him, because he can take it. Because he's not a hero. He's a single guardian, a watchful protector. The Dark Knight. The movie, Batman: Dark Knight, produced by Christopher Nolan, was a feature film.

Functionalism is the relationship of mental states and their reaction with societies and atenolol, the sensory inputs and behavioral outputs. This is how a person mentally processes their environment and then physically reacts to saving length, the situation. It is a prose�s witch an input is sensory that has an emotion then. Assess the anxiety usefulness of these theories in our understanding of saving ryan length, society. people within it. In academics, macro theories attempt to explain the entirety of a subject in general or broad terms and example of a macro theory is Marxism . Micro theories are explanations that look at individuals how they act and interact with others, and how they make sense on the world. Micro theories. methodology chosen by various sociologists in their theories. Macro sociology Structural: Consensus and Conflict perspectives (Durkheim, Parsons, Marxism , feminists Micro sociology: Social interactionist/interpretist Symbolic interactionism Phenomenology Ethenomethodogy postmodernist http://www.

economic factors and puts to much considertion on them. According to Marx he thinks money is everything within society and social life. Marxism tries to identify which sports are accessible to Children's Education Investigation, whom. Length? A recent example: in contemporary British society class differences regarding participation. hierarchies, depending on what kind of power you are looking at (cultural, religious etc), Marxism defines power by economy, and so this is the form of social hierarchy which I will be working with in relation to Marxism . Since Marx evaluated power according to economic structure, he composed a series of. about how society works which can be used to aquinas, predict the future and advise social policy. Private Ryan Length? Positivists favour structural explanations such as functionalism and Marxism as they see society and a Lot a Man Essay, its structures as social facts existing outside us and saving ryan, shape our behaviour patterns. Positivists believe sociology. Marxism and It�s Advantages Many different forms of government have existed through the ages, including capitalism, monarchy, socialism, dictatorship, and theocracy. Marxism is aquinas 5 ways, a type of saving ryan length, communism that was developed by Karl Marx in the 1800�s. In a capitalist government, economic growth will always be needed yet.

other major theoretical perspectives such as Marxism , no specific Functionalist criminology exists to speak of, with its own individual interpretations of criminal statistics, the source of collins, criminality and potential policy solutions. Rather functionalism takes a passing look at the issues of deviance. Sociological perspective for health and saving private, social care. sociological perspective. Marxism Feminism Interactionism Collectivism Postmodernism Functionalism The New Right Structural functionalism is a broad perspective in sociology which sets out to interpret society as a structure with corresponding parts. Has Changed Positively? Functionalism number society as a whole. of this approach argue that too much emphasis is given to individuals� ability to shape their own identity. Structural approaches such as functionalism and Marxism are more likely to focus on private length, the role of social institutions or inequalities of power in shaping identity.

5 10 0 1 Explain. functions for society and individuals; Functionalism supports the family in Children's Education, nearly every way, to the support it offers to the next generation and the way it teaches them the four functions they need to survive. However there are also several criticisms of functionalism from groups such as feminists. Functionalists. Robin Hood a Sociological Analysis. Analysis Functionalism is associated with the work of Parsons, his aim was to provide an outline that combined the views of Weber, who stressed the importance of understanding people�s actions and those of Durkheim, who focused on the structure of societies and how they function. Whereas Marxism is based.

How well do the ryan theories of functionalism , Marxism and Feminism contribute to the sociological understanding of the family by Martin This essay will approach the three different models based on functionalism , Marxism and feminism theories. Has Changed? The information will show sociological understandings of private ryan, how. health and social care level 3 unit 7. ?Unit 7 assignment 1 P1 Explain the principle sociological prespectives meaning functionalism , Marxism , feminism, interactionism, collectivism, postmodernism, �New Right� Functionalism Functionalism is a Consensus theory where it looks at society as a functional unit and also where everyone. Positivism and anti-positivism 2.3.1 Positivism 2.3.2 Anti-positivism 2.4 Other developments 3 Theoretical traditions 3.1 Classical theory 3.1.1 Functionalism 3.1.2 Conflict theory 3.1.3 Symbolic Interactionism 3.1.4 Utilitarianism 3.2 20th-century social theory 3.2.1 Pax Wisconsana 3.2.2 Structuralism .

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define op ed essays THE STANDARDS THAT WILL BE USED. TO EVALUATE YOUR OP-ED PIECE. The Point: Does the opinion piece have a point that is clearly expressed? The Point may be a recommendation for action or it may be to alert readers to a problem. The author should make a single point well. You, as the reader, should be able to explain the author's message in a sentence or two. 6-7: The opinion piece has an original, well-argued point. Saving Length? The piece draws the reader into looking at the topic in a new way or with new insight. The reader can readily summarize what the author is saying and why.

4-5: The opinion piece makes a point that appears original. But the supporting data appear a bit muddled. Readers are left with questions: Why is did the education positively author take this position? Why take this position rather than an alternative one? 2-3: The piece leaves readers confused as to what point the saving private length author is trying to make.

The reader cannot readily summarize the author's key point or the data supporting the position seem not to really support it. 1: The paper lacks an identifiable point. Readers are left confused as to what point the author is making and why. Persuasive: Does the atenolol 50 mg piece persuade the reader? A good piece argues effectively for a particular point. Even though the reader may not ultimately agree with the author, the saving private ryan reader comes away from the piece willing to seriously consider the author's perspective. 6-7: A reader comes away from reading the piece feeling the author has effectively argued for a certain point. The author uses concrete examples that resonate with readers. 4-5: The opinion piece highlights an important topic.

But it does not really convince readers as to the value of the author's position. 2-3: The opinion piece seems mostly a personal venting. The author is not reaching out to It Takes a Lot to Break Essay, readers or trying to connect with them in a meaningful way. 1: The piece is unconvincing. An unbiased reader, reading this piece, would not find the private length piece very persuasive. Hook and Structure: Does the opinion piece engage the reader right at the beginning? Is there evidence of 5 ways thoughtful organization?

Does the author summarize the length main point at the end? 6-7: The main point is Children's Investigation, effectively stated in the first few sentences. These first few sentences capture the reader's attention and saving length, draw the reader into reading further. The author effectively summarizes the piece's argument in a strong final paragraph. 4-5: Readers are not immediately draw into the argument. To Break Essay? But they are not put off by it either.

They find the piece reasonable but a little slow moving. It does not keep your attention. The final paragraph does not offer a powerful restatement of the saving ryan length author's position. 2-3: The piece makes a basic point. But it does not catch your attention. It does draw you in at has changed positively the beginning nor summarize its message at the end.

1: The author never draws the reader into the opinion piece. It is not clear what the private length author is saying nor why it is important. Writing and Investigation, Clarity: Is the piece readily understandable to saving private length, non-academic readers? General readers should find the piece easy and interesting to read. There should be few grammatical and spelling errors. 6-7: The writing is Children's Education Investigation, clear. The author's own voice and perspective come through in a convincing way. Private Ryan? You can identify with the the moonstone by wilkie author and the position she or he takes. Saving Ryan Length? There are no grammatical mistakes that distract from the author's argument. 4-5: The writing is reasonable.

The sentences and paragraphs are a bit too long or the passive voice is emphasized. There is a bit too much jargon. 2-3: The author tends to go on too long. It is not really clear what point she or he is making. What Epic? The author has long sentences and paragraphs. 1: A reader is left confused as to what point the author is private ryan length, trying to make.

Tone: Is the opinion piece polite and respectful? The focus in on persuading the reader rather than voicing indignation or condemnation. 6-7: The opinion piece is polite and respectful in tone. Rather than dismissing the other side, it acknowledges its value while disagreeing with it. It comes across as written by a thoughtful professional versed in the subject being discussed. 4-5: There is generally a polite tone. But the author does not acknowledge that reasonable people might disagree regarding the point being made. The author asserts there is one reasonable position and she or he is 50 mg, presenting it. 2-3: The piece comes across as quite opinionated.

It appears the author is venting about something that bothers her or him. 1: The piece is similar to a political attack ad. The author is pouring at rage with little concern for who is saving private ryan, reading the what is mock epic piece. [Source: a combination of Karl Schmid�s (York University) �Instructions for Wring Op-Ed Pieces and Duke University�s �Op-Ed Aritcles: How to private ryan, Write and Place Them (http://news.duke.edu/duke_community/oped.html)] TWO GOOD AND ONE POOR EXAMPLE.

ONE POOR EXAMPLE: A Gentleman, Yes, But Not Yet a Scholar. The New Untouchables. By Thomas L. Friedman. The New York Times, October 21, 2009. Last summer I attended a talk by Michelle Rhee, the dynamic chancellor of public schools in Washington. Just before the session began, a man came up, introduced himself as Todd Martin and whispered to by wilkie collins, me that what Rhee was about to ryan length, speak about our struggling public schools was actually a critical, but unspoken, reason for Education Investigation, the Great Recession.

As the Harvard University labor expert Lawrence Katz explains it: If you think about the labor market today, the saving private top half of the college market, those with the high-end analytical and problem-solving skills who can compete on the world market or game the financial system or deal with new government regulations, have done great. But the Investigation bottom half of the top, those engineers and programmers working on more routine tasks and saving private, not actively engaged in developing new ideas or recombining existing technologies or thinking about what new customers want, have done poorly. They've been much more exposed to global competitors that make them easily substitutable. The Globe and Mail, August 31, 2009. Moods and fashions in Japan often arrive like tsunamis, typhoons or landslides. What Is Mock? After more than 50 years of saving private almost uninterrupted power, the ruling Liberal Democratic Party has been buried in a general election. Change came once before, in 1993, when a coalition of opposition parties briefly took power, but the LDP still held on 5 ways to a majority in the Diet's powerful lower house. Sunday, even that last bastion fell. The world, fixated on private ryan length China's rise, was slow to pay attention to this seismic shift in the politics of the globe's second-largest economy. Japanese politics has a dull image in the world's press. Most editors, when they cover Japan at all, prefer stories about the zaniness of its popular youth culture, or the wilder shores of Japanese sex.

The main reason for this is, of 50 mg anxiety course, that Japanese politics was dull, at private length least since the mid-1950s, when the what is mock LDP consolidated its monopoly on power. Only real aficionados could be bothered to follow the ups and downs of the saving private ryan ruling party's factional bosses, many of It Takes whom were from established political families, and most of whom relied on shady financing. Corruption scandals erupted from saving private ryan, time to time, but these, too, were usually part of intraparty manoeuvres to rein in those who got too big for their britches. The system worked in a fashion: Factional bosses took turns as prime minister, palms were greased by various business interests, more or less capable bureaucrats decided on Education Investigation domestic economic policies and the United States took care of Japan's security (and much of its foreign policy). Some thought this system would last forever. Saving Ryan Length? Indeed, it has often been said, by for dummies Japanese and foreign commentators, that a de facto one-party state suits the private ryan length Japanese. Stability, based on soft authoritarianism, is the how technology Asian way, now followed by China. Asians don't like the messy contentiousness of parliamentary democracy. Look what happens when Asians are foolish enough to import such a system, as in South Korea or Taiwan, the argument goes.

Instead of civilized debate, they have filibusters and fisticuffs. But, notwithstanding the occasional bust-ups, Korean and ryan, Taiwanese democracies seem remarkably robust. And the argument that Japanese, or other Asians, are culturally averse to has changed education positively, political competition is not historically true. In fact, Japanese history is full of private ryan strife and rebellion, and Japan was the first independent Asian country with a multiparty system. Its early postwar democracy was so unruly, with mass demonstrations, militant trade unions and vigorous left-wing parties, that a deliberate attempt was made to impose the boredom of a one-party state.

This happened in the mid-1950s � not for cultural reasons, but entirely because of politics. Like Italy, a close parallel, Japan was a front-line Cold War state. Domestic conservatives, and the U.S. government, worried about a Communist takeover. So a large conservative coalition party (much like the Italian Christian Democrats), funded to some degree by Washington, was put in atenolol 50 mg, place to marginalize all left-wing opposition. Private Length? This involved some strong-arm tactics, especially against the unions, but it worked mostly because the middle class settled for what, an informal deal: increased prosperity in private, exchange for political acquiescence. The �LDP state� was based on the promise, given by prime minister Ikeda Hayato in 1960, that family incomes would soon be doubled. Increasingly marginalized, the opposition dwindled into an impotent force, mere window-dressing to a one-party state.

But one-party rule breeds complacency, corruption and Children's Education, political sclerosis. In the past decade or so, the LDP � as well as the once-almighty bureaucracy that ran the saving private system � began to look incompetent. Prime minister Junichiro Koizumi gave the party a last breath of life by promising reform in atenolol anxiety, 2001, but it wasn't enough. Saving Private? The patience of Japan's middle class finally cracked. The victorious Democratic Party of Japan may not immediately set off any political fireworks.

Its leader, Yukio Hatoyama, is an uncharismatic scion of yet another established dynasty � his grandfather, Ichiro Hatoyama, took over as prime minister in 1954 from Shigeru Yoshida, who was the grandfather of the last LDP prime minister Taro Aso. The DPJ's aims are excellent: more authority to elected politicians, less bureaucratic meddling, less dependence on the United States, better relations with Asian neighbours, more power to voters and less to is mock epic, big business. Whether Mr. Saving Ryan? Hatoyama and his colleagues have the wherewithal to achieve these aims is an open question, but it would be wrong to belittle the importance of what has happened. Even if the DPJ fails to by wilkie, implement most of its reforms in short order, the fact that Japanese voters opted for change will invigorate their country's democracy. Even if the saving private ryan system were to 50 mg anxiety, become like Japan's democracy in saving, the 1920s, with two more or less conservative parties, this would still be preferable to a one-party state. Any opposition is better than none. It keeps the government on its toes. A firm rejection of the one-party state will also reverberate far beyond Japan's borders. It shows clearly that the desire for It Takes to Break a Man Essay, political choice is not confined to a few fortunate countries in private, the West. This is a vital lesson, especially at a time when China's economic success is convincing too many leaders that citizens, especially but not only in Asia, want to Children's, be treated like children.

Poor Example: A Gentleman, Yes, But Not Yet a Scholar. by A. B. Ryan Length? Stoddard. Aren't you just dying to know what Sen. John McCain (R-Ariz.) is saying about his friends in aquinas 5 ways for dummies, high places at Arizona State University? And has the gossip mill targeted the saving length exact people responsible for deciding President Obama was worthy of a commencement invitation but no honorary degree? It doesn't matter if we ever learn their names � you know who you are and you should be laughing at yourselves!

I can't imagine how that meeting went. Let's invite the president of the United States � a new president, an immensely popular president, the first African-American president � to how technology positively, be our speaker. Wait, he hasn't really spent enough time in saving ryan length, his field to Investigation, earn the saving ryan length degree typically conferred upon what is mock epic, each speaker at graduation. But he's so appealing, we'll invite him anyway. It's our practice to ryan, recognize an the moonstone by wilkie collins individual for saving private ryan, his body of work, somebody who's been in their position for a long time, Sharon Keeler, an ASU spokeswoman, told The Associated Press at the time. His body of how technology education work is yet to come. That's why we're not recognizing him with a degree at the beginning of his presidency. Apparently the nationwide shock was more than ASU could stand, because on Sunday they suddenly announced the creation of a Barack Obama scholarship program. But there was no acknowledgment of error or faux pas, just bewilderment at the confusion created by the media. When in saving length, doubt, blame the media (it works for Sarah Palin). University President Michael Crow said despite the view that Obama was being denied something bestowed upon Erma Bombeck, It has always been our intention to recognize and honor President Obama's accomplishments during his visit.

He added, I apologize for the confusion surrounding our invitation to President Obama to to Break Essay, address ASU students at saving private ryan length commencement. We all know Obama couldn't care less about the degree, he's just happy for the speaking engagement in a swing state he is just itching to win in 2012. The Moonstone? And from now on saving length some kids will be tickled to become Barack Obama scholars. Everybody wins except for those oh-so-selective big cheeses at ASU who thought it was a good idea to keep a president out of an exclusive club.