A Brief Guide to Algebraic Number Theory

✍ Scribed by H. P. F. Swinnerton-Dyer

• Publisher: Cambridge University Press
• Year: 2001
• Tongue: English
• Leaves: 154
• Series: London Mathematical Society Student Texts
• Edition: annotated edition
• Category: Library

✦ Synopsis

This account of Algebraic Number Theory is written primarily for beginning graduate students in pure mathematics, and encompasses everything that most such students are likely to need; others who need the material will also find it accessible. It assumes no prior knowledge of the subject, but a firm basis in the theory of field extensions at an undergraduate level is required, and an appendix covers other prerequisites. The book covers the two basic methods of approaching Algebraic Number Theory, using ideals and valuations, and includes material on the most usual kinds of algebraic number field, the functional equation of the zeta function and a substantial digression on the classical approach to Fermat's Last Theorem, as well as a comprehensive account of class field theory. Many exercises and an annotated reading list are also included.

✦ Table of Contents

Cover ......Page 1
Content ......Page 4
Preface page ......Page 6
1 The ring of integers ......Page 9
2 Ideals and factorization ......Page 17
3 Embedding in the complex numbers ......Page 26
4 Change of fields ......Page 31
5 Normal extensions ......Page 34
6 Valuations and completions ......Page 39
7 Field extensions and ramification ......Page 48
8 The Different ......Page 51
9 Ideles and Adeles ......Page 56
10 Quadratic fields ......Page 63
11 Pure cubic fields ......Page 70
12 Biquadratic fields ......Page 71
13 Cyclotomic fields ......Page 73
13.1 Class numbers of cyclotomic fields ......Page 76
13.2 Fermat's Last Theorem ......Page 81
14 Zeta functions and L-series ......Page 87
15 Analytic continuation and the functional equation ......Page 92
16 Density theorems ......Page 102
17 The classical theory ......Page 106
18 Chevalley's reformulation ......Page 111
19 Reciprocity theorems ......Page 114
20 The Kronecker-Weber Theorem ......Page 120
Al.1 Finitely generated abelian groups and lattices ......Page 125
Al.2 Norms and Traces ......Page 129
A1.3 Haar measure ......Page 130
A2 Additional topics ......Page 132
A2.1 Characters and duality ......Page 133
A2.2 Fourier transforms ......Page 136
A2.3 Galois theory for infinite extensions ......Page 140
Exercises ......Page 143
Suggested further reading ......Page 151
Index ......Page 153