Algebraic Number Theory

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Paperback | 5 April 1993

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372 Pages

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22.86 x 15.24 x 2.11

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This book originates from graduate courses given in Cambridge and London. It provides a brisk, thorough treatment of the foundations of algebraic number theory, and builds on that to introduce more advanced ideas. Throughout, the authors emphasise the systematic development of techniques for the explicit calculation of the basic invariants, such as rings of integers, class groups, and units. Moreover they combine, at each stage of development, theory with explicit computations and applications, and provide motivation in terms of classical number-theoretic problems. A number of special topics are included that can be treated at this level but can usually only be found in research monographs or original papers, for instance: module theory of Dedekind domains; tame and wild ramifications; Gauss series and Gauss periods; binary quadratic forms; and Brauer relations. This is the only textbook at this level which combines clean, modern algebraic techniques together with a substantial arithmetic content. It will be indispensable for all practising and would-be algebraic number theorists.

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'It promises to become a classic.' Monatshefte fur Mathematik '... very nice and carefully written ... many excellent exercises...' European Mathematical Society Newsletter

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Preface	p. ix
Notation	p. xiii
Introduction	p. 1
Algebraic Foundations	p. 8
Fields and Algebras	p. 8
Integrality and Noetherian properties	p. 26
Dedekind Domains	p. 35
Algebraic Theory	p. 35
Valuations and absolute values	p. 58
Completions	p. 70
Module theory over a Dedekind domain	p. 87
Extensions	p. 102
Decomposition and ramification	p. 102
Discriminants and differents	p. 120
Non-ramified and tamely ramified extensions	p. 132
Ramification in Galois extensions	p. 142
Classgroups and Units	p. 152
Elementary results	p. 152
Lattices in Euclidean space	p. 156
Classgroups	p. 164
Units	p. 168
Fields of low degree	p. 175
Quadratic fields	p. 175
Biquadratic fields	p. 193
Cubic and sextic fields	p. 198
Cyclotomic Fields	p. 205
Basic theory	p. 205
Characters	p. 213
Quadratic fields revisited	p. 220
Gauss sums	p. 231
Elliptic curves	p. 241
Diophantine Equations	p. 251
Fermat's last theorem	p. 251
Quadratic forms	p. 254
Cubic equations	p. 269
L-functions	p. 277
Dirichlet series	p. 277
The Dedekind zeta-function	p. 283
Dirichlet L-functions	p. 295
Primes in an arithmetic progression	p. 297
Evaluation of L(1, x) and explicit class number formulae for cyclotomic fields	p. 299
Quadratic fields, yet again	p. 306
Brauer relations	p. 309
Characters of Finite Abelian Groups	p. 327
Exercises	p. 335
Suggested Further Reading	p. 349
Glossary of Theorems	p. 352
Index	p. 353
Table of Contents provided by Syndetics. All Rights Reserved.

Algebraic Number Theory

At a Glance

Paperback

Industry Reviews

Shipping

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Defective items

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