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Algebraic Number Theory : Cambridge Studies in Advanced Mathematics (Paperback) - A. Frohlich

Algebraic Number Theory

Cambridge Studies in Advanced Mathematics (Paperback)

By: A. Frohlich, M. J. Taylor, B. Bollobas (Editor), W. Fulton (Editor), A. Katok (Editor)

Paperback

Published: 5th April 1993
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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.

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

Prefacep. ix
Notationp. xiii
Introductionp. 1
Algebraic Foundationsp. 8
Fields and Algebrasp. 8
Integrality and Noetherian propertiesp. 26
Dedekind Domainsp. 35
Algebraic Theoryp. 35
Valuations and absolute valuesp. 58
Completionsp. 70
Module theory over a Dedekind domainp. 87
Extensionsp. 102
Decomposition and ramificationp. 102
Discriminants and differentsp. 120
Non-ramified and tamely ramified extensionsp. 132
Ramification in Galois extensionsp. 142
Classgroups and Unitsp. 152
Elementary resultsp. 152
Lattices in Euclidean spacep. 156
Classgroupsp. 164
Unitsp. 168
Fields of low degreep. 175
Quadratic fieldsp. 175
Biquadratic fieldsp. 193
Cubic and sextic fieldsp. 198
Cyclotomic Fieldsp. 205
Basic theoryp. 205
Charactersp. 213
Quadratic fields revisitedp. 220
Gauss sumsp. 231
Elliptic curvesp. 241
Diophantine Equationsp. 251
Fermat's last theoremp. 251
Quadratic formsp. 254
Cubic equationsp. 269
L-functionsp. 277
Dirichlet seriesp. 277
The Dedekind zeta-functionp. 283
Dirichlet L-functionsp. 295
Primes in an arithmetic progressionp. 297
Evaluation of L(1, x) and explicit class number formulae for cyclotomic fieldsp. 299
Quadratic fields, yet againp. 306
Brauer relationsp. 309
Characters of Finite Abelian Groupsp. 327
Exercisesp. 335
Suggested Further Readingp. 349
Glossary of Theoremsp. 352
Indexp. 353
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521438346
ISBN-10: 0521438349
Series: Cambridge Studies in Advanced Mathematics (Paperback)
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 372
Published: 5th April 1993
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.78 x 15.19  x 2.08
Weight (kg): 0.52