Reciprocity Laws. From Euler to Eisenstein.

Table of Contents

1. The Genesis of Quadratic Reciprocity

1.1. P. Fermat
1.2. L. Euler
1.3. A.-M. Legendre
1.4. C.-F. Gauss

2. Quadratic Number Fields

2.1. Quadratic Fields
2.2. Genus Theory
2.3. Genus Characters
2.4. The Lucas-Lehmer Test
2.5. Hilbert Symbols and K₂

3. Cyclotomic Number Fields

3.1. Cyclotomic Fields
3.2. Primality Tests
3.3. Quadratic Gauss Sums
3.4. Cyclotomic Units

4. Power Residues and Gauss Sums

4.1. Residue Symbols in Number Fields
4.2. Gauss's Lemma
4.3. Discriminants
4.4. Kummer Extensions
4.5. Characters of Abelian Groups
4.6. Sums of Gauss, Jacobi and Eisenstein

5. Rational Reciprocity Laws

5.1. L. Dirichlet
5.2. A. Scholz
5.3. E. Lehmer
5.4. Rational Quartic Reciprocity
5.5. Residue Characters of Quadratic Units

6. Quartic Reciprocity

6.1. Splitting of Primes
6.2. Quartic Gauss and Jacobsthal Sums
6.3. The Quartic Reciprocity Law
6.4. Applications
6.5. Quartic Reciprocity in some Quartic Fields

7. Cubic Reciprocity

7.1. Splitting of Primes
7.2. The Cubic Reciprocity Law
7.3. Sextic Reciprocity
7.4. Cubic Reciprocity in some Quartic Fields

8. Eisenstein's Analytic Proofs

8.1. Quadratic Reciprocity
8.2. Abel's Construction of Elliptic Functions
8.3. Elliptic Functions
8.4. Quartic and Cubic Reciprocity
8.5. Quadratic Reciprocity in Quadratic Fields
8.6. Kronecker's Jugendtraum
8.7. The Determination of Gauss Sums

9. Octic Reciprocity

9.1. The Rational Octic Reciprocity Law
9.2. Eisenstein's Reciprocity Law
9.3. Elliptic Gauss Sums
9.4. The Octic Reciprocity Law
9.5. Scholz's Octic Reciprocity Law

10. Gauss's Last Entry

10.1. Connections with Quartic Reciprocity
10.2. Counting Points with Cyclotomic Numbers
10.3. Counting Points with Jacobi Sums
10.4. The Classical Zeta Functions
10.5. Counting Points with Zeta Functions

11. Eisenstein Reciprocity

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11.1. Factorization of Gauss Sums
11.2. Eisenstein Reciprocity for l-th Powers
11.3. The Stickelberger Congruence
11.4. Class Groups of Abelian Number Fields

A. Dramatis Personae

B. Chronology of Proofs

C. Some Open Problems

References

Author Index

Subject Index