Lemmermeyer F. Reciprocity Laws: From Euler to Eisenstein. 2000.
Lemmermeyer F. Reciprocity Laws: From Euler to Eisenstein. - Springer, 2000. - 514 p.
This book is about the development of reciprocity laws, starting from conjectures of Euler and discussing the contributions of Legendre, Gauss, Dirichlet, Jacobi, and Eisenstein. Readers knowledgeable in basic algebraic number theory and Galois theory will find detailed discussions of the reciprocity laws for quadratic, cubic, quartic, sextic and octic residues, rational reciprocity laws, and Eisenstein's reciprocity law. An extensive bibliography will be of interest to readers interested in the history of reciprocity laws or in the current research in this area.
1. The Genesis of Quadratic Reciprocity
2. Quadratic Number Fields
3. Cyclotomic Number Fields
4. Power Residues and Gauss Sums
5. Rational Reciprocity Laws
6. Quartic Reciprocity
7. Cubic Reciprocity
8. Eisenstein's Analytic Proofs
9. Octic Reciprocity
10. Gauss's Last Entry
11. Eisenstein Reciprocity
A. Dramatis Personae
B. Chronology of Proofs
C. Some Open Problems