1. To the Gauss Reciprocity Law. - 1. 1. Basic Facts. - 1. 2. Modules in ?. - 1. 3. Euclidean Algorithm and Continued Fractions. - 1. 4. Continued-Fraction Expansion of Irrational Numbers. - 1. 5. Concept of Groups. - 1. 6. Subgroups and Quotient Groups. - 1. 7. Ideals and Quotient Rings. - 1. 8. Isomorphisms and Homomorphisms. - 1. 9. Polynomial Rings. - 1. 10. Primitive Roots. - 1. 11. Algebraic Integers. - 1. 12. Characters of Abelian Groups. - 1. 13. The Gauss Reciprocity Law. - 2. Basic Concepts of Algebraic Number Fields. - 2. 1. Field Extensions. - 2. 2. Galois Theory. - 2. 3. Norm Trace and Discriminant. - 2. 4. Gauss Sum and Jacobi Sum. - 2. 5. Construction of a Regular l-gon. - 2. 6. Subfields of the lth Cyclotomic Field. - 2. 7. Cohomology of Cyclic Groups. - 2. 8. Finite Fields. - 2. 9. Ring of Integers Ideals and Discriminant. - 2. 10. Fundamental Theorem of Ideal Theory. - 2. 11. Residue Class Rings. - 2. 12. Decomposition of Primes in Number Fields. - 2. 13. Discriminant and Ramification. - 2. 14. Hilbert Theory. - 2. 15. Artin Map. - 2. 16. Artin Maps of Subfields of the lth Cyclotomic Field. - 2. 17. The Artin Map in Quadratic Fields. - 3. Analytic Methods. - 3. 1 Lattices in ?n. - 3. 2. Minkowski's Theorem. - 3. 3. Dirichlet's Unit Theorem. - 3. 4. Pre-Zeta Functions. - 3. 5. Dedekind Zeta Function. - 3. 6. The mth Cyclotomic Field. - 3. 7. Dirichlet L-Functions. - 3. 8. Dirichlet's Theorem on Arithmetical Progressions. - 4. The lth Cyclotomic Field and Quadratic Fields. - 4. 1. Determination of Gauss Sums. - 4. 2. L-Functions and Gauss Sums. - 4. 3. Class Numbers of Subfields of the lth Cyclotomic Field. - 4. 4. Class Number of ?$$(\sqrt {{l^*}} )$$. - 4. 5. Ideal Class Groups of Quadratic Fields. - 4. 6. Cohomology of Quadratic Fields. - 4. 7. Gauss Genus Theory. - 4. 8. Quadratic Irrationals. - 4. 9. Real Quadratic Fields and Continued Fractions. - Answers and Hints to Exercises. - Notes. - A. Peano Axioms. - B. Fundamental Theorem of Algebra. - C. Zorn's Lemma. - D. Quadratic Fields and Quadratic Forms. - List of Mathematicians. - Comments onthe Bibliography. Language: English