A Brief Guide to Algebraic Number Theory

by H. P. F. Swinnerton-Dyer

• ISBN13: 9780521802925
• ISBN10: 052180292X
• Format: Hardcover
• Copyright: 2001-03-19
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface	vii
Numbers and Ideals	1	(30)
The ring of integers	1	(8)
Ideals and factorization	9	(9)
Embedding in the complex numbers	18	(5)
Change of fields	23	(3)
Normal extensions	26	(5)
Valuations	31	(24)
Valuations and completions	31	(9)
Field extensions and ramification	40	(3)
The Different	43	(5)
Ideles and Adeles	48	(7)
Special fields	55	(24)
Quadratic fields	55	(7)
Pure cubic fields	62	(1)
Biquadratic fields	63	(2)
Cyclotomic fields	65	(14)
Class numbers of cyclotomic fields	68	(5)
Fermat's Last Theorem	73	(6)
Analytic methods	79	(19)
Zeta functions and L-series	79	(5)
Analytic continuation and the functional equation	84	(10)
Density theorems	94	(4)
Class Field Theory	98	(19)
The classical theory	98	(5)
Chevalley's reformulation	103	(3)
Reciprocity theorems	106	(6)
The Kronecker-Weber Theorem	112	(5)
Appendix	117	(18)
A1 Prerequisites	117	(7)
A1.1 Finitely generated abelian groups and lattices	117	(4)
A1.2 Norms and Traces	121	(1)
A1.3 Haar measure	122	(2)
A2 Additional topics	124	(11)
A2.1 Characters and duality	125	(3)
A2.2 Fourier transforms	128	(4)
A2.3 Galois theory for infinite extensions	132	(3)
Exercises	135	(8)
Suggested further reading	143	(2)
Index	145