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