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Table of Contents
1.
Historical Introduction,
Page 1
Part 1.
Algebraic Number Theory
2.
The ring of integers of a number field,
Page 9
2-1.
Integers,
Page 9
2-2.
Ideal theory,
Page 13
3.
The locally compact groups of number theory,
Page 20
3-1.
Completions,
Page 20
3-2.
Adeles,
Page 32
3-3.
Ideles,
Page 38
4.
Local fields,
Page 46
4-1.
The topology of local fields,
Page 46
4-2.
Completions of number fields are local fields,
Page 52
4-3.
The arithmetic of local fields,
Page 55
4-4.
Local ramification theory,
Page 62
4-5.
Local Hilbert theory,
Page 67
5.
Global arithmetic,
Page 72
5-1.
Splitting of primes,
Page 72
5-2.
The valuations of an algebraic number field,
Page 79
5-3.
Global ramification theory,
Page 80
5-4.
Galois actions,
Page 82
5-5.
Global Hilbert theory,
Page 84
5-6.
An example—quadratic fields,
Page 88
6.
Applications,
Page 93
6-1.
Special functions,
Page 93
6-2.
Cyclotomic extensions,
Page 96
Part 2.
Harmonic Analysis
7.
Harmonic analysis on locally compact abelian groups,
Page 111
7-1.
Duality theory,
Page 111
7-2.
Fourier transforms,
Page 121
7-3.
The Schwartz-Bruhat space,
Page 127
Part 3.
Hecke's L-functions
8.
Hecke's L-series,
Page 133
8-1.
Local zeta functions,
Page 133
8-2.
Global zeta functions,
Page 145
8-3.
Hecke L-series,
Page 156
9.
Applications of Hecke L-functions,
Page 162
9-1.
Splitting of pries,
Page 162
9-2.
Abelian L-functions,
Page 164
9-3.
Tchebotarev's density theorem
Page 168
9-4.
Dirichlet's L-functions,
Page 171
9-5.
Class number formulas,
Page 172
Part 4.
Class Field Theory
10.
Class field theory,
Page 181
10-1.
Abelian L-functions are Hecke L-functions,
Page 182
11.
The inequalities of class field theory,
Page 188
11-1.
The second inequality,
Page 188
11-2.
The first inequality,
Page 197
12.
Class field theory,
Page 206
12-1.
Proof of the reciprocity law,
Page 206
12-2.
Local class field theory,
Page 218
12-3.
Global class field theory,
Page 226
13.
Applications,
Page 238
13-1.
Kronecker's theorem,
Page 238
13-2.
The power reciprocity law,
Page 241
Part 5.
The Prime Number Theorem
14.
The prime number theorem,
Page 251
14-1.
Natural densities,
Page 251
14-2.
Analytic preliminaries,
Page 257
14-3.
The prime number theorem,
Page 260
Suggestions for further reading,
Page 273
Appendix: algebraic preliminaries,
Page 275
Bibliography,
Page 277
Index,
Page 279
Edition Notes
Bibliography: p. 277-278.
Classifications
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