Algebraic Theory of Numbers Translated from the French by Allan J. Silberger

by Samuel, Pierre

• ISBN13: 9780486466668
• ISBN10: 0486466663
• Format: Paperback
• Copyright: 2008-05-19
• Publisher: Dover Publications

Summary

Algebraic number theory introduces students to new algebraic notions as well as related concepts: groups, rings, fields, ideals, quotient rings, and quotient fields. This text covers the basics, from divisibility theory in principal ideal domains to the unit theorem, finiteness of the class number, and Hilbert ramification theory. 1970 edition.

Table of Contents

Translator's Introduction	p. 7
Introduction	p. 9
Notations, Definitions, and Prerequisites	p. 11
Principal ideal rings	p. 13
Divisibility in principal ideal rings	p. 13
An example: the diophantine equations X[superscript 2] + Y[superscript 2] = Z[superscript 2] and X[superscript 4] + Y[superscript 4] = Z[superscript 4]	p. 15
Some lemmas concerning ideals; Euler's [characters not reproducible]-function	p. 17
Some preliminaries concerning modules	p. 19
Modules over principal ideal rings	p. 21
Roots of unity in a field	p. 23
Finite fields	p. 23
Elements integral over a ring; elements algebraic over a field	p. 27
Elements integral over a ring	p. 27
Integrally closed rings	p. 30
Elements algebraic over a field. Algebraic extensions	p. 30
Conjugate elements, conjugate fields	p. 32
Integers in quadratic fields	p. 34
Norms and traces	p. 36
The discriminant	p. 38
The terminology of number fields	p. 41
Cyclotomic fields	p. 42
The field of complex numbers is algebraically closed	p. 44
Noetherian rings and Dedekind rings	p. 46
Noetherian rings and modules	p. 46
An application concerning integral elements	p. 47
Some preliminaries concerning ideals	p. 47
Dedekind rings	p. 49
The norm of an ideal	p. 52
Ideal classes and the unit theorem	p. 53
Preliminaries concerning discrete subgroups of R[superscript n]	p. 53
The canonical imbedding of a number field	p. 56
Finiteness of the ideal class group	p. 57
The unit theorem	p. 59
Units in imaginary quadratic fields	p. 62
Units in real quadratic fields	p. 62
A generalization of the unit theorem	p. 64
The calculation of a volume	p. 66
The splitting of prime ideals in an extension field	p. 68
Preliminaries concerning rings of fractions	p. 68
The splitting of a prime ideal in an extension	p. 70
The discriminant and ramification	p. 73
The splitting of a prime number in a quadratic field	p. 76
The quadratic reciprocity law	p. 77
The two-squares theorem	p. 81
The four-squares theorem	p. 82
Galois extensions of number fields	p. 86
Galois theory	p. 86
The decomposition and inertia groups	p. 89
The number field case. The Frobenius automorphism	p. 91
An application to cyclotomic fields	p. 92
Another proof of the quadratic reciprocity law	p. 92
A Supplement, Without Proofs	p. 94
Exercises	p. 97
Bibliography	p. 106
Index	p. 108
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