Notes on class field theory

Preface Preface

Last modified: January 17, 2023.

This text is a lightly edited version of the lecture notes of a course on class field theory (Math 254B) that I gave at UC Berkeley in the spring of 2002. To describe the scope of the course, I can do no better than to quote from the original syllabus:

Class field theory, the study of abelian extensions of number fields, was a crowning achievement of number theory in the first half of the 20th century. It brings together, in a unified fashion, the quadratic and higher reciprocity laws of Gauss, Legendre et al, and vastly generalizes them. Some of its consequences (e.g., the Chebotaryov density theorem) apply even to nonabelian extensions.

Our approach in this course will be to begin with the formulations of the statements of class field theory, omitting the proofs (except for the Kronecker-Weber theorem, which we prove first). We then proceed to study the cohomology of groups, an important technical tool both for class field theory and for many other applications in number theory. From there, we set up a local form of class field theory, then proceed to the main results.

The assumed background for the course was a one-semester graduate course in algebraic number theory, including the following topics: number fields and rings of integers; structure of the class and unit groups; splitting, ramification, and inertia of prime ideals under finite extensions; different and discriminant; basic properties of local fields. In fact, most of the students in Math 254B had attended such a course that I gave the previous semester (Math 254A) based on chapters I, II, and III of Neukirch's book [37]; for that reason, it was natural to use that book as a primary reference. However, no special features of that presentation are assumed, so just about any graduate-level text on algebraic number theory (e.g., Fröhlich-Taylor [11], Janusz [25], Jarvis [26], Lang [33]) should provide suitable background.

After the course ended, I kept the lecture notes posted on my web site in their originally written, totally uncorrected state. Despite their roughness, I heard back from many people over the years who had found them useful; in response to this, I decided to prepare a corrected version of the notes. This project gained some steam when I had the opportunity to teach another class¹ on class field theory in winter 2021. This occasioned some more significant revisions than I had previously dared, including a small degree of rearrangement of the material; however, I have tried to retain most of the original structure, on the grounds that the informality of the original notes contributed to their readibility. In other words, this document is not intended as a standalone replacement for a good book on class field theory!

I maintain very few claims of originality concerning the presentation of the material. Besides [37], the main source of inspiration was Milne's lecture notes on class field theory [36] (and by extension the original development by Artin and Tate [1]). The basic approach may be summarized as follows: I follow Milne's treatment of local class field theory using group cohomology, then follow Neukirch to recast local class field theory in the style of Artin-Tate's class formations, then reuse the same framework to obtain global class field theory. Since the original draft of these notes was written, several treatments have appeared in a similar vein: [38], [17]. (See also [13] for a modern exposition of a more classical approach.)

My winter 2021 class, having taken place during the COVID-19 pandemic, was given online with recorded lectures. The recordings continue to be available from the course web site².

Thanks to Zonglin Jiang, Justin Lacini, and Zongze Liu for their feedback on previous drafts, and to the participants in Math 204B (winter 2021) for “test-driving” the HTML version and generating much additional feedback. Thanks also to Rob Beezer and David Farmer for their assistance with the conversion from LaTeX to PreTeXt³, which made it feasible to produce an HTML version⁴ in sync with the PDF version⁵.