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Table of contents for Reciprocity laws : from Euler to Eisenstein / Franz Lemmermeyer.
Contents
Preface v
1. The Genesis of Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . 1
1.1 P. Fermat 1
1.2 L. Euler 3
1.3 A.-M. Legendre 6
1.4 C.-F. Gauss 9
2. Quadratic Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1 Quadratic Fields 43
2.2 Genus Theory 47
2.3 Genus Characters 52
2.4 The Lucas-Lehmer Test 56
2.5 Hilbert Symbols and KZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3. Cyclotomic Number Fields 79
3.1 Cyclotomic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Primality Tests 85
3.3 Quadratic Gauss Sums . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4 Cyclotomic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4. Power Residues and Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.1 Residue Symbols in Number Fields 111
4.2 Gauss's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3 Discriminants . 116
4.4 Kummer Extensions 119
4.5 Characters of Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.6 Sums of Gauss, Jacobi and Eisenstein . . . . . . . . . . . . . . . . . . . . . 126
5. Rational Reciprocity Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.1 L. Dirichlet 154
5.2 A. Scholz 160
5.3 E. Lehmer 161
5.4 Rational Quartic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.5 Residue Characters of Quadratic Units . . . . . . . . . . . . . . . . . . . . 168
6. Quartic Reciprocity 185
6.1 Splitting of Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.2 Quartic Gauss and Jacobsthal Sums . . . . . . . . . . . . . . . . . . . . . . 190
6.3 The Quartic Reciprocity Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.5 Quartic Reciprocity in some Quartic Fields . . . . . . . . . . . . . . . . 198
7. Cubic Reciprocity 209
7.1 Splitting of Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.2 The Cubic Reciprocity Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.3 Sextic Reciprocity 217
7.4 Cubic Reciprocity in some Quartic Fields . . . . . . . . . . . . . . . . . . 220
8. Eisenstein's Analytic Proofs 235
8.1 Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.2 Abel's Construction of Elliptic Functions . . . . . . . . . . . . . . . . . . 239
8.3 Elliptic Functions . . . _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.4 Quartic and Cubic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.5 Quadratic Reciprocity in Quadratic Fields . . . . . . . . . . . . . . . . . 256
8.6 Kronecker's Jugendtraum . 260
8.7 The Determination of Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . 265
9. Octic Reciprocity 289
9.1 The Rational Octic Reciprocity Law . . . . . . . . . . . . . . . . . . . . . . 289
9.2 Eisenstein's Reciprocity Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
9.3 Elliptic Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
9.4 The Octic Reciprocity Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
9.5 Scholz's Octic Reciprocity Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
10. Gauss's Last Entry 317
10.1 Connections with Quartic Reciprocity . . . . . . . . . . . . . . . . . . . . . 317
10.2 Counting Points with Cyclotomic Numbers . . . . . . . . . . . . . . . . 321
10.3 Counting Points with Jacobi Sums . . . . . . . . . . . . . . . . . . . . . . . . 326
10.4 The Classical Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
10.5 Counting Points with Zeta Functions . . . . . . . . . . . . . . . . . . . . . . 335
11. Eisenstein Reciprocity 357
11.1 Factorization of Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
11.2 Eisenstein Reciprocity for P-th Powers . . . . . . . . . . . . . . . . . . . . . 361
11.3 The Stickelberger Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
11.4 Class Groups of Abelian Number Fields . . . . . . . . . . . . . . . . . . . 371
A. Dramatis Personae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
B. Chronology of Proofs 413
C. Some Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
References 419
Author Index 472
Subject Index ................................................ 484