Modern number theory, according to Hecke, dates from Gauss's quadratic reciprocity law. The various extensions of this law and the generalizations of the domains of study for number theory have led to a rich network of ideas, which has had effects throughout mathematics, in particular in algebra. This volume of the Encyclopaedia presents the main structures and results of algebraic number theory with emphasis on algebraic number fields and class field theory. Koch has written for the non-specialist. He assumes that the reader has a general understanding of modern algebra and elementary number theory. Mostly only the general properties of algebraic number fields and related structures are included. Special results appear only as examples which illustrate general features of the theory. A part of algebraic number theory serves as a basic science for other parts of mathematics, such as arithmetic algebraic geometry and the theory of modular forms. For this reason, the chapters on basic number theory, class field theory and Galois cohomology contain more detail than the others. This book is suitable for graduate students and research mathematicians who wish to become acquainted with the main ideas and methods of algebraic number theory.
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Book Description
Editorial Reviews
Review
From the reviews: "... The author succeeded in an excellent way to describe the various points of view under which Class Field Theory can be seen. ... In any case the author succeeded to write a very readable book on these difficult themes." Monatshefte fuer Mathematik, 1994 "... Number theory is not easy and quite technical at several places, as the author is able to show in his technically good exposition. The amount of difficult material well exposed gives a survey of quite a lot of good solid classical number theory... Conclusion: for people not already familiar with this field this book is not so easy to read, but for the specialist in number theory this is a useful description of (classical) algebraic number theory." Medelingen van het wiskundig genootschap, 1995
--This text refers to an alternate
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Language Notes
Text: English (translation)
Original Language: Russian --This text refers to an alternate Hardcover edition.
Original Language: Russian --This text refers to an alternate Hardcover edition.
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local analytic manifolds, their coefficient rings, nonassociate prime elements, mth cyclotomic field, decomposable forms, prime cyclotomic fields, units with norm, primitive quadratic character, fundamental parallelepiped, nonsingular quadratic form, divisor classes, decomposition into prime factors, rational quadratic forms, integral divisors, rational integer coefficients, reduced module, principal genus, principal divisors, rational integers, unit divisor, complex isomorphisms, prime divisors, irregular primes, primitive mth root, minimum polynomial Key Phrases - Capitalized Phrases (CAPs): (learn more)
New York, Use Problem, Hence Theorem, Modern Algebra, Full Module Browse Sample Pages:
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local analytic manifolds, their coefficient rings, nonassociate prime elements, mth cyclotomic field, decomposable forms, prime cyclotomic fields, units with norm, primitive quadratic character, fundamental parallelepiped, nonsingular quadratic form, divisor classes, decomposition into prime factors, rational quadratic forms, integral divisors, rational integer coefficients, reduced module, principal genus, principal divisors, rational integers, unit divisor, complex isomorphisms, prime divisors, irregular primes, primitive mth root, minimum polynomial Key Phrases - Capitalized Phrases (CAPs): (learn more)
New York, Use Problem, Hence Theorem, Modern Algebra, Full Module Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
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100
books
cite this book:
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- Algebraic Number Theory by H. Koch on 7 pages
- Elementary and Analytic Theory of Algebraic Numbers (Springer Monographs in Mathematics) by Wladyslaw Narkiewicz on 4 pages
- A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) by Kenneth Ireland on page 338, Back Matter (1), and Back Matter (2)
- Number Fields (Universitext) by Daniel A. Marcus on page 6, page 141, and Back Matter
- Algebra IV: Infinite Groups. Linear Groups (Encyclopaedia of Mathematical Sciences) (v. 4) by A.I. Kostrikin in Back Matter (1), and Back Matter (2)
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