Items related to An Introduction to Algebraic Number Theory
Synopsis
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 on the Bibliography.
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