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1. Algebraic Number Theory - Wikipedia, The Free Encyclopedia
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished
http://en.wikipedia.org/wiki/Algebraic_number_theory
##### Algebraic number theory
From Wikipedia, the free encyclopedia Jump to: navigation search Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers . This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K Q , and studying their algebraic properties such as factorization , the behaviour of ideals , and field extensions. In this setting, the familiar features of the integers —such as unique factorization —need not hold. The virtue of the primary machinery employed— Galois theory group cohomology group representations , and L -functions —is that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers.
##### Contents
• Basic notions
##### edit Unique factorization and the ideal class group
One of the first properties of Z that can fail in the ring of integers O of an algebraic number field K is that of the unique factorization of integers into prime numbers . The prime numbers in Z are generalized to irreducible elements in O , and though the unique factorization of elements of O into irreducible elements may hold in some cases (such as for the Gaussian integers Z [i]), it may also fail, as in the case of

2. Algebraic Number Theory - On Opentopia, Find Out More About Algebraic Number The
Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic number s which are root s of polynomial s with rational
http://encycl.opentopia.com/term/Algebraic_number_theory
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##### Algebraic number theory
Encyclopedia A AL ALG : Algebraic number theory
Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic number s which are root s of polynomial s with rational coefficients. An algebraic number field is any finite (and therefore algebraic field extension of the rational numbers. These domains contain elements analogous to the integer s, the so-called algebraic integer Galois theory group cohomology class field theory ... group representation s and L-function Many number theoretic questions are best attacked by studying them modulo p for all primes p (see finite field s). This is called localization and it leads to the construction of the p-adic number s; this field of study is called local analysis and it arises from algebraic number theory.
From Wikipedia , the Free Encyclopedia. Original article here . Support Wikipedia by contributing or donating
All text is available under the terms of the GNU Free Documentation License See for details.

 3. Algebraic Number Theory. (Open Library) Algebraic number theory by Jack Schwartz, 1961,New York University Institute of Mathematical Sciences edition, in Englishhttp://openlibrary.org/books/OL16585478M/Algebraic_number_theory.

4. Algebraic Number Theory - Wikivisual
This article or section does not cite its references or sources. You can help Wikipedia by introducing appropriate citations. This article has been tagged since October 2006.
http://en.wikivisual.com/index.php/Algebraic_number_theory
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##### Algebraic number theory
You can help Wikipedia by introducing appropriate citations. This article has been tagged since October 2006 Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. An algebraic number field is any finite (and therefore algebraic field extension of the rational numbers. These domains contain elements analogous to the integers , the so-called algebraic integers . In this setting, the familiar features of the integers (e.g. unique factorization Galois theory group cohomology class field theory ... group representations and L-functions Many number theoretic questions are best attacked by studying them modulo p for all primes p (see finite fields ). This is called localization and it leads to the construction of the p -adic numbers ; this field of study is called local analysis and it arises from algebraic number theory.

 5. Algebraic Number Theory | Ask.com Encyclopedia Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering ahttp://www.ask.com/wiki/Algebraic_number_theory?qsrc=3044

 6. Algebraic Number Theory In English - Dictionary And Translation algebraic number theory. Dictionary terms for algebraic number theory in English, English definition for algebraic number theory, Thesaurus and Translations of algebraic numberhttp://www.babylon.com/definition/algebraic_number_theory/English

 7. Algebraic Number Theory In - Dictionary And Translation Algebraic Number Theory. Dictionary terms for Algebraic Number Theory, definition for Algebraic Number Theory, Thesaurus and Translations of Algebraic Number Theory to Chinesehttp://www.babylon.com/definition/Algebraic_Number_Theory/

 8. Algebraic Number Theory By Richard Mollin Get the lowest price on Algebraic Number Theory (2nd Edition) by Richard Mollin. Read customer reviews and compare prices at more than 40 online bookstores before you buyhttp://www.allbookstores.com/book/9781439845981/Algebraic_Number_Theory.html

 9. Algebraic Number Theory - Discussion And Encyclopedia Article. Who Is Algebraic Algebraic number theory. Discussion about Algebraic number theory. Ecyclopedia or dictionary article about Algebraic number theory.http://www.knowledgerush.com/kr/encyclopedia/Algebraic_number_theory/

10. Algebraic Number Theory OpenCourseWare: A Free Graduate Study Course By MIT On T
Jan 3, 2009 Topics in Algebraic Number Theory is a free OpenCourseWare offered by MIT. The first in a series of courses on algebraic number theory,
http://educhoices.org/articles/Algebraic_Number_Theory_OpenCourseWare_A_Free_Gra
'Topics in Algebraic Number Theory' is a free OpenCourseWare offered by MIT. The first in a series of courses on algebraic number theory, this course covers a wide... Algebraic Number Theory OpenCourseWare: A Free Graduate Study Course by MIT on the Study... Graduate Level Math and... OpenCourseWare (Graduate...
##### Algebraic Number Theory OpenCourseWare: A Free Graduate Study Course by MIT on the Study of Computer Algebra
Published Jan 03, 2009 RSS Feed
##### Topics in Algebraic Number Theory: Course Specifics
Degree Level Free Audio Video Downloads Graduate Yes No No Yes Lectures/Notes Study Materials Tests/Quizzes No Yes Yes
##### Topics in Algebraic Number Theory: Course Description
Professor Kiran Kedlaya's course, 'Topics in Algebraic Number Theory,' provides a solid foundation for further study of number theory. Presented as a series of 25 lectures, the course guides students through a variety of topics, including Dirichlet's units theorem, decomposition and inertia groups. Other topics covered will include algebraic valuations, basic analytic methods, ramification and basic class field theory. While covering these foundational topics, students will also employ computer algebra to complete numerical experiments using SAGE (Software for Algebra and Geometry Exploration). Numerical experimentation is the key to discovering results in number theory, explains Professor Kedlaya. Several texts are recommended, including Gerald J. Janusz's

11. MP473 2000
A course by Keith Matthews. Lecture notes (GIF slides), bibliography, web resources.
http://www.numbertheory.org/courses/MP473/
##### Prepared by Keith Matthews
Email: webmaster@numbertheory.org
Web: http://www.numbertheory.org/keith.html
##### Study Suggestions
Students are encouraged in tutorials to raise any difficulties encountered with the problem sheets and lecture material. Students should try to keep up to date with study of their lectures, so as to be able to understand subsequent lectures. They are also urged to do as many problems as possible. By doing problems, students will soon discover their strong and weak points. The lecture notes will contain enough explanations and examples to make the definitions, theorems and arguments clear. However some students will need further examples and explanations of certain points and I recommend they peruse books from the reading list below. Most of these books have lots of examples and develop the concepts in greater detail than we have time for in our short course of lectures.
##### Course Outline
The course is an introduction to algebraic number theory, especially quadratic and cyclotomic fields.

12. Lecture Notes Of Advanced Courses
Lecture Notes of Advanced Courses Notes from some of my graduate courses, courtesy of Jeff Achter Algebraic Number Theory, Part I (Math 620, fall 1993) dvi file (.dvi format) or
http://www.math.upenn.edu/~chai/coursenotes.html
##### Lecture Notes of Advanced Courses
Notes from some of my graduate courses, courtesy of Jeff Achter

13. A Course In Algebraic Number Theory
http://www.math.uiuc.edu/~r-ash/ANT.html
##### A Course In Algebraic Number Theory

 14. Algebraic Number Theory File Format PDF/Adobe Acrobathttp://www.jmilne.org/math/CourseNotes/ANT.pdf

15. PlanetMath: Algebraic Number Theory
Mar 15, 2005 This entry is a collection of links to entries on algebraic number theory in Planetmath (therefore bound to be always under construction).
http://planetmath.org/encyclopedia/AlgebraicNumberTheory.html
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talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About algebraic number theory (Topic)
##### Algebraic Number Theory
This entry is a collection of links to entries on algebraic number theory in Planetmath (therefore bound to be always under construction ). It is the hope of the author(s) that someday this can be used as a ``graduate text'' to learn the subject by reading the individual entries listed here. Each section contains a brief description of the concepts, which is expanded in the entries. Some of the concepts might be missing in Planetmath as of today (please consider writing an entry on them!). In order to organize the entry in sections , we followed the main reference Mar
##### Introduction
The entry number theory contains a nice introduction to the broad subject. From very early on, mathematicians have tried to understand the

 16. Algebraic Number Theory: Definition From Answers.com The study of properties of real numbers, especially integers, using the methods of abstract algebra.http://www.answers.com/topic/algebraic-number-theory

17. 11R: Algebraic Number Theory: Global Fields
Jan 14, 2000 There is also a mailing list ALGEBRAICNUMBER-THEORY ; information is available at the Algebraic Number Theory Preprint Archive
http://www.math.niu.edu/~rusin/known-math/index/11RXX.html
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##### Applications and related fields
For complex multiplication, See 11G15 The rings of integers in these fields are also studied in Commutative Ring Theory , where we may find discussion of Unique Factorization Domains (UFDs), Euclidean domains, etc.
##### Subfields
• Algebraic numbers; rings of algebraic integers
• PV-numbers and generalizations; other special algebraic numbers
• Polynomials (irreducibility, etc.)
• Cubic and quartic extensions
• Cyclotomic extensions
• Other abelian and metabelian extensions
• Other number fields
• Iwasawa theory
• Units and factorization
• Class numbers, class groups, discriminants
• Galois theory
• Integral representations related to algebraic numbers; Galois module structure of rings of integers, See Also 20C10
• Class field theory

18. Algebraic Number Theory -- From Wolfram MathWorld
Oct 11, 2010 Algebraic number theory is the branch of number theory that deals with algebraic numbers. Historically, algebraic number theory developed as
http://mathworld.wolfram.com/AlgebraicNumberTheory.html
 Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Terr Algebraic Number Theory Algebraic number theory is the branch of number theory that deals with algebraic numbers . Historically, algebraic number theory developed as a set of tools for solving problems in elementary number theory , namely Diophantine equations (i.e., equations whose solutions are integers or rational numbers ). Using algebraic number theory, some of these equations can be solved by " lifting " from the field of rational numbers to an algebraic extension of More recently, algebraic number theory has developed into the abstract study of algebraic numbers and number fields themselves, as well as their properties. SEE ALSO: Algebraic Extension Algebraic Integer Algebraic Number Class Group ... Number Theory This entry contributed by David Terr REFERENCES: Stewart, I. and Tall, D. Algebraic Number Theory and Fermat's Last Theorem, 3rd ed. Wellesley, MA: A K Peters, 2000. CITE THIS AS: Terr, David . "Algebraic Number Theory." From MathWorld A Wolfram Web Resource, created by

19. Algebraic Number Theory
Algebraic Number Theory From the review The present book has as its aim to resolve a discrepancy in the textbook literature and to provide a
http://www.springer.com/mathematics/numbers/book/978-3-540-65399-8