Kato Kazuya, Kurokawa Nobushige, Saito Takeshi. Number Theory 2: Introduction to Class Field Theory

Transl. from the Japan.: Masato Kuwata, Katsumi Nomizu. — American Mathematical Society, 2011. — viii, 242 p. — (Translations of Mathematical Monographs. Vol. 240). — ISBN 978-0-8218-1355-3.This book, the second of three related volumes on number theory, is the English translation of the original Japanese book. Here, the idea of class field theory, a highlight in algebraic number theory, is first described with many concrete examples. A detailed account of proofs is thoroughly exposited in the final chapter. The authors also explain the local-global method in number theory, including the use of ideles and adeles. Basic properties of zeta and L-functions are established and used to prove the prime number theorem and the Dirichlet theorem on prime numbers in arithmetic progressions. With this book, the reader can enjoy the beauty of numbers and obtain fundamental knowledge of modern number theory.Preface to the English Edition
What is Class Field Theory?
Examples of class field theoretic phenomena
Cyclotomic fields and quadratic fields
An outline of class field theory
Summary
Exercises
Local and Global Fields
A curious analogy between numbers and functions
Places and local fields
Places and field extension
Adele rings and idele groups
Summary
Exercises
ζ (II)
The emergence of ζ
Riemann ζ and Dirichlet L
Prime number theorems
The case of F_p[T] 130
Dedekind ζ and Hecke L
Generalization of the prime number theorem
Summary
Exercises
Class Field Theory (II)
The content of class field theory
Skew fields over a global or local field
Proof of the class field theory
Summary
Exercises
Appendix B. Galois Theory
Galois theory
Normal and separable extensions
Norm and trance
Finite fields
Infinite Galois theory
Appendix C. Lights of Places
Hensel’s lemma
The Hasse principleAnswers to Questions
Answers to Exercises
IndexTrue PDF