Algebraic theory of numbers

Front Cover
Title
Contents
Translator's Introduction
Introduction
Notations, definitions, and prerequisites
I. Principal ideal rings
1.1. Divisibility in principal ideal rings
1.2. An example: the diophantine equations X^2 + Y^2 = Z^2 and X^4 + Y^4 = Z^4
1.3. Some lemmas concerning ideals; Euler’s phi-function
1.4. Some preliminaries concerning modules
1.5. Modules over principal ideal rings
1.6. Roots of unity in a field
1.7. Finite fields
II. Elements integral over a ring; elements algebraic over a field
2.1. Elements integral over a ring
2.2. Integrally closed rings
2.3. Elements algebraic over a field. Algebraic extensions
2.4. Conjugate elements, conjugate fields
2.5. Integers in quadratic fields
2.6. Norms and traces
2.7. The discriminant
2.8. The terminology of number fields
2.9. Cyclotomic fields
Appendix: The field C of complex numbers is algebraically closed
III. Noetherian rings and Dedekind rings
3.1. Noetherian rings and modules
3.2. An application concerning integral elements
3.3. Some preliminaries concerning ideals
3.4. Dedekind rings
3.5 The norm of an ideal
IV. Ideal classes and the unit theorem
4.1. Preliminaries concerning discrete subgroups of R^n
4.2. The canonical imbedding of a number field
4.3. Finiteness of the ideal class group
4.4. The unit theorem
4.5. Units in imaginary quadratic fields
4.6. Units in real quadratic fields
4.7. A generalization of the unit theorem
Appendix: The calculation of a volume
V. The splitting of prime ideals in an extension field
5.1. Preliminaries concerning rings of fractions
5.2. The splitting of a prime ideal in an extension
5.3. The discriminant and ramification
5.4. The splitting of a prime number in a quadratic field
5.5. The quadratic reciprocity law
5.6. The two squares theorem
5.7. The four squares theorem
VI. Galois extensions of number fields
6.1. Galois theory
6.2. The decomposition and inertia groups
6.3. The number field case. The Frobenius automorphism
6.4. An application to cyclotomic fields
6.5. Another proof of the quadratic reciprocity law
A supplement, without proofs
Exercises
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Review Problems
Bibliography
Index
Back Matter