An Introduction to Algebraic Number Theory

This book is a translation of my book Suron Josetsu (An Introduction to Number Theory), Second Edition, published by Shokabo, Tokyo, in 1988. The translation is faithful to the original globally but, taking advantage of my being the translator of my own book, I felt completely free to reform or deform the original locally everywhere. When I sent T. Tamagawa a copy of the First Edition of the original work two years ago, he immediately pointed out that I had skipped the discussion of the class numbers of real quadratic fields in terms of continued fractions and (in a letter dated 2/15/87) sketched his idea of treating continued fractions without writing explicitly continued fractions, an approach he had first presented in his number theory lectures at Yale some years ago. Although I did not follow his approach exactly, I added to this translation a section (Section 4. 9), which nevertheless fills the gap pointed out by Tamagawa. With this addition, the present book covers at least T. Takagi's Shoto Seisuron Kogi (Lectures on Elementary Number Theory), First Edition (Kyoritsu, 1931), which, in turn, covered at least Dirichlet's Vorlesungen. It is customary to assume basic concepts of algebra (up to, say, Galois theory) in writing a textbook of algebraic number theory. But I feel a little strange if I assume Galois theory and prove Gauss quadratic reciprocity.

Contenu

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.