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Class Field Theory

Emil Artin and John Tate
American Mathematical Society
Publication Date: 
Number of Pages: 
AMS/Chelsea Publishing 366
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Fernando Q. Gouvêa
, on

This new edition of the famous Artin-Tate notes on class field theory is a must-have, even for those who already have a copy of the original. It's fairly easy to explain why. For one thing, this is a classic, a book that has inspired a generation of number theorists. It has long been out of print and hard to get, however.

But the big news is that the new edition has been entirely re-set using TeX. Tate has taken advantage of this to update and correct the notes, adding footnotes, references, and two new mathematical sections: one sketching the analytic proof of the “second inequality” and another on Weil groups. This has, of course, made a very good book even better.

Most of us, I think, would not want to start our study of class field theory with this book, but aspiring number theorists will want to get to it at some point. It's hard going but deep, insightful, and essential.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.



  • Preliminaries
  • The first fundamental inequality
  • Second fundamental inequality
  • Reciprocity law
  • The existence theorem
  • Connected component of idèle classes
  • The Grunwald-Wang theorem
  • Higher ramification theory
  • Explicit reciprocity laws
  • Group extensions
  • Abstract class field theory
  • Weil groups
  • Bibliography