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A Brief Guide to Algebraic Number Theory : London Mathematical Society Student Texts - H. P. F. Swinnerton-Dyer

A Brief Guide to Algebraic Number Theory

London Mathematical Society Student Texts

Hardcover

Published: 14th January 2011
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This is an account of Algebraic Number Theory, a field which has grown to touch many other areas of pure mathematics. It is written primarily for beginning graduate students in pure mathematics, and encompasses everything that most such students are likely to need; others who need the material will also find it accessible. It assumes no prior knowledge of the subject, but a firm basis in the theory of field extensions at an undergraduate level is required, and an appendix covers other prerequisites. The book covers the two basic methods of approaching Algebraic Number Theory, using ideals and valuations, and includes material on the most usual kinds of algebraic number field, the functional equation of the zeta function and a substantial digression on the classical approach to Fermat's Last Theorem, as well as a comprehensive account of class field theory. Many exercises and an annotated reading list are also included.

'... masterfully written. It has to be recommended to number theorists and more general to working algebraists.' J. Schoissengeier, Monatshefte fur Mathematik '... ideally suited for non-specialists who would like to know quickly what algebraic number theory is about but who do not wish to study thick volumes dealing with that subject.' Wladyslaw Narkiewicz, Zentralblatt fur Mathematik '... very readable. It is suitable for graduate students or non-specialists who wish to learn algebraic number theory.' Gabor Megyesi, Acta Sci. Math. 'It is written in a nice (but necessarily concise) style, containing exercises to broaden the reader's knowledge and skills. The result is a very solid introduction to the subject.' EMS 'The carefully chosen material is further accompanied by a large number of illustrative examples and exercises. Those features make the book an excellent guide for the reader who wants to find out what algebraic number theory is all about.' Nieuw Archief voor Wiskunde

Prefacep. vii
Numbers and Idealsp. 1
The ring of integersp. 1
Ideals and factorizationp. 9
Embedding in the complex numbersp. 18
Change of fieldsp. 23
Normal extensionsp. 26
Valuationsp. 31
Valuations and completionsp. 31
Field extensions and ramificationp. 40
The Differentp. 43
Ideles and Adelesp. 48
Special fieldsp. 55
Quadratic fieldsp. 55
Pure cubic fieldsp. 62
Biquadratic fieldsp. 63
Cyclotomic fieldsp. 65
Class numbers of cyclotomic fieldsp. 68
Fermat's Last Theoremp. 73
Analytic methodsp. 79
Zeta functions and L-seriesp. 79
Analytic continuation and the functional equationp. 84
Density theoremsp. 94
Class Field Theoryp. 98
The classical theoryp. 98
Chevalley's reformulationp. 103
Reciprocity theoremsp. 106
The Kronecker-Weber Theoremp. 112
Appendixp. 117
Prerequisitesp. 117
Finitely generated abelian groups and latticesp. 117
Norms and Tracesp. 121
Haar measurep. 122
Additional topicsp. 124
Characters and dualityp. 125
Fourier transformsp. 128
Galois theory for infinite extensionsp. 132
Exercisesp. 135
Suggested further readingp. 143
Indexp. 145
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521802925
ISBN-10: 052180292X
Series: London Mathematical Society Student Texts
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 160
Published: 14th January 2011
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 1.3
Weight (kg): 0.41