Algebraic Theory of Numbers

Translated from the French by Allan J. Silberger

By: PIERRE SAMUEL

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Paperback | 19 May 2008

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Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics - algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Galois theory, Noetherian rings and modules, and rings of fractions. It covers the basics, starting with the divisibility theory in principal ideal domains and ending with the unit theorem, finiteness of the class number, and the more elementary theorems of Hilbert ramification theory. Numerous examples, applications, and exercises appear throughout the text.

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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
Table of Contents provided by Ingram. All Rights Reserved.

Algebraic Theory of Numbers

Translated from the French by Allan J. Silberger

At a Glance

Paperback

Shipping

How to return your order

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