Kato (U. of Chicago), Kurokawa (Tokyo Institute of Technology), and Saito (U. of Tokyo) explore similarities between algebraic number fields and algebraic function fields in one variable over finite fields, explain adele rings and idele groups, derive several prime number theorems, and prove the main theorem of class field theory. Originally published in Japanese by Iwanami Shoten in 1998. The English version of volume one was published by AMS in 1999. Annotation ©2011 Book News, Inc., Portland, OR (booknews.com)