Kato K., Kurokawa N., Saito T. Number theory 2. Introduction to Class Field Theory

Transl. from the Japan.: Masato Kuwata, Katsumi Nomizu. — Providence: 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. The translation of the first volume was published as Number Theory 1: Fermat's Dream, Translations of Mathematical Monographs (Iwanami Series in Modern Mathematics), vol. 186, American Mathematical Society, 2000.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 principle.Answers to Questions.
Answers to Exercises.
Index