Reciprocity laws: from Euler to Eisenstein [1 ed.] 9783540669579, 3540669574
This book is about the development of reciprocity laws, starting from conjectures of Euler and discussing the contributi
182 33 4MB
English Pages 514 Year 2000
Table of contents :
Preface
Contents
I: The genesis fo quadratic reciprocity
1. P. Fermat
2. L. Euler
3. A. M. Legendre
4. C. F. Gauss
II: Quadratic number fields
1. Quadratic fields
2. Genus theory
3. Genus characters
4. The Lucas-Lehmer test
5. Hilbert symbols and K2
III: Cyclotomic number fields
1. Cyclotomic fields
2. Primality tests
3. Quadratic Gauss sums
4. Cyclotomic units
IV: Power residues and Gauss sums
1. Residue symbols in number fields
2. Gauss's lemma
3. Discriminants
4. Kummer extensions
5. Characters of Abelian groups
6. Sums of Guass, Jacobi and Eisenstein
V: Rational reciprocity laws
1. L. Dirichlet
2. A. Scholz
3. E. Lehmer
4. Rational quartic reciprocity
5. Residue characters of quadratic units
VI: Quartic reciprocity
1. Splitting of primes
2. Quartic Gauss and Jacobsthal sums
3. The quartic reciprocity law
4. Applications
5. Quartic reciprocity in some quartic fields
VII Cubic reciprocity
1. Splitting of primes
2. The cubic reciprocity law
3. Sextic reciprocity
4. Cubic reciprocity in some quartic fields
VIII: Eisenstein's analytic proofs
1. Quadratic reciprocity
2. Abel's construction of elliptic functions
3. Elliptic funcitons
4. Quartic and cubic reciprocity
5. Quadratic reciprocity in quadratic fields
6. Kronecker's Jugendtraum
7. The determination of Gauss sums
IX: Octic reciprocity
1. The rational octic reciprocity law
2. Eisenstein's reciprocity law
3. Elliptic Gauss sums
4. The octic reciprocity law
5. Sholz's octic reciprocity law
X: Gauss's last entry
1. Connections with quartic reciprocity
2. Counting points with cyclotomic numbers
3. Counting points with Jacobi sums
4. The classical zeta functions
5. Counting points with zeta functions
XI: Eisenstein reciprocity
1. Factorization of Gauss sums
2. Eisenstein reciprocity for ell-th powers
3. The Stickelberg congruence
4. Class groups of abelian number fields
A. Dramatis personae
B. Chronology of proofs
C. Some open problems
References
Preface
Contents
I: The genesis fo quadratic reciprocity
1. P. Fermat
2. L. Euler
3. A. M. Legendre
4. C. F. Gauss
II: Quadratic number fields
1. Quadratic fields
2. Genus theory
3. Genus characters
4. The Lucas-Lehmer test
5. Hilbert symbols and K2
III: Cyclotomic number fields
1. Cyclotomic fields
2. Primality tests
3. Quadratic Gauss sums
4. Cyclotomic units
IV: Power residues and Gauss sums
1. Residue symbols in number fields
2. Gauss's lemma
3. Discriminants
4. Kummer extensions
5. Characters of Abelian groups
6. Sums of Guass, Jacobi and Eisenstein
V: Rational reciprocity laws
1. L. Dirichlet
2. A. Scholz
3. E. Lehmer
4. Rational quartic reciprocity
5. Residue characters of quadratic units
VI: Quartic reciprocity
1. Splitting of primes
2. Quartic Gauss and Jacobsthal sums
3. The quartic reciprocity law
4. Applications
5. Quartic reciprocity in some quartic fields
VII Cubic reciprocity
1. Splitting of primes
2. The cubic reciprocity law
3. Sextic reciprocity
4. Cubic reciprocity in some quartic fields
VIII: Eisenstein's analytic proofs
1. Quadratic reciprocity
2. Abel's construction of elliptic functions
3. Elliptic funcitons
4. Quartic and cubic reciprocity
5. Quadratic reciprocity in quadratic fields
6. Kronecker's Jugendtraum
7. The determination of Gauss sums
IX: Octic reciprocity
1. The rational octic reciprocity law
2. Eisenstein's reciprocity law
3. Elliptic Gauss sums
4. The octic reciprocity law
5. Sholz's octic reciprocity law
X: Gauss's last entry
1. Connections with quartic reciprocity
2. Counting points with cyclotomic numbers
3. Counting points with Jacobi sums
4. The classical zeta functions
5. Counting points with zeta functions
XI: Eisenstein reciprocity
1. Factorization of Gauss sums
2. Eisenstein reciprocity for ell-th powers
3. The Stickelberg congruence
4. Class groups of abelian number fields
A. Dramatis personae
B. Chronology of proofs
C. Some open problems
References
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