Algebraic Number Theory

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Cambridge University Press, Feb 4, 1993 - Mathematics - 355 pages
2 Reviews
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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Contents

Algebraic Foundations
8
I2 Integrality and Noetherian properties
26
Dedekind Domains
35
II2 Valuations and absolute values
58
II3 Completions
70
II4 Module theory over a Dedekind domain
87
Extensions
102
III2 Discriminants and differents
120
VI3 Quadratic fields revisited
220
VI4 Gauss sums
231
VI5 Elliptic curves
241
Diophantine Equations
251
VII2 Quadratic forms
254
VII3 Cubic equations
269
Lfunctions
277
VIII2 The Dedekind zeta function
283

III3 Nonramified and tamely ramified extensions
132
III4 Ramification in Galois extensions
142
Classgroups and Units
152
IV2 Lattices in Euclidean space
156
IV3 Classgroups
164
IV4 Units
168
Fields of low degree
175
V2 Biquadratic fields
193
V3 Cubic and sextic fields
198
Cyclotomic Fields
205
VI2 Characters
213
VIII3 Dirichlet Lfunctions
295
VIII4 Primes in an arithmetic progression
297
VIII5 Evaluation of L1𝜒 and explicit class number formulae for cyclotomic fields
299
VIII6 Quadratic fields yet again
306
VIII7 Brauer relations
309
Characters of Finite Abelian Groups
327
Exercises
335
Suggested Further Reading
349
Glossary of Theorems
352
Index
353
Copyright

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Page xiii - C denote the natural numbers, the integers, the rational numbers, the real numbers, the complex numbers respectively.
Page 1 - We begin by considering the classical problem of when the prime number p can be represented as the sum of the squares of two integers.

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About the author (1993)

Frohlich, Imperial College-London, and Robinson College-Cambridge.

Taylor, UMIST, Manchester, England.

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