CONTENTS
Notation and Conventions = xi
1. To the Gauss Reciprocity Law = 1
1.1. Basic Facts = 2
1.2. Modules in Z = 4
1.3. Euclidean Algorithm and Continued Fractions = 8
1.4. Continued-Fraction Expansion of Irrational Numbers = 12
1.5. Concept of Groups = 16
1.6. Subgroups and Quotient Groups = 21
1.7. Ideals and Quotient Rings = 23
1.8. Isomorphisms and Homomorphisms = 25
1.9. Polynomial Rings = 28
1.10. Primitive Roots = 30
1.11. Algebraic Integers = 34
1.12. Characters of Abelian Groups = 37
1.13. The Gauss Reciprocity Law = 41
2. Basic Concepts of Algebraic Number Fields = 44
2.1. Field Extensions = 44
2.2. Galois Theory = 48
2.3. Norm, Trace, and Discriminant = 53
2.4. Gauss Sum and Jacobi Sum = 55
2.5. Construction of a Regular l-gon = 58
2.6. Subfields of the lth Cyclotomic Field = 60
2.7. Cohomology of Cyclic Groups = 63
2.8. Finite Fields = 68
2.9. Ring of Integers, Ideals, and Discriminant = 69
2.10. Fundamental Theorem of Ideal Theory = 74
2.11. Residue Class Rings = 78
2.12. Decomposition of Primes in Number Fields = 81
2.13. Discriminant and Ramification = 86
2.14. Hilbert Theory = 89
2.15. Artin Map = 93
2.16. Artin Maps of Subfields of the lth Cyclotomic Field = 97
2.17. The Artin Map in Quadratic Fields = l00
3. Analytic Methods = 105
3.1 Lattices in Rn = 105
3.2. Minkowski's Theorem = 109
3.3. Dirichlet's Unit Theorem = 113
3.4. Pre-Zeta Functions = 119
3.5. Dedekind Zeta Function = 123
3.6. The mth Cyclotomic Field = 132
3.7. Dirichlet L-Functions = 134
3.8. Dirichlet's Theorem on Arithmetical Progressions = 140
4. The Ith Cyclotomic Fietd and Quadratic Fields = 143
4.1. Determination of Gauss Sums = 144
4.2. L-Functions and Gauss Sums = 149
4.3. Class Numbers of Subfields of the lth Cyclotomic Field = 152
4.4. Class Number of Q( l* ) = 155
4.5. Ideal Class Groups of Quadratic Fields = 159
4.6. Cohomology of Quadratic Fields = 166
4.7. Gauss Genus Theory = 173
4.8. Quadratic Irrationals = 179
4.9. Real Quadratic Fields and Continued Fractions = 186
Answers and Hints to Exercises = 195
Notes = 210
A. Peano Axioms = 210
B. Fundamental Theorem of Algebra = 211
C. Zorn's Lemma = 212
D. Quadratic Fields and Quadratic Forms = 212
List of Mathematicians = 215
Bibliography = 216
Comments on the Bibliography = 220
Index = 211
