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Class Field Theory

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

[Reviewed by
Fernando Q. Gouvêa
, on

“The question before us, folks, is whether it is possible to write an accessible introduction to Class Field Theory.”

“That would seem to depend on the meaning of ‘accessible,’ wouldn’t it? After all, without some background in algebraic number theory, Galois theory, and so on, one can’t even understand the statements of the theorems.”

“Well, of course. So let’s posit a fairly well-prepared class. Say, graduate students who have taken their intro courses, so they have a fairly firm grasp of complex analysis, Galois Theory, and even a little topology. And let’s say they have just finished their course in algebraic number theory, so studying class field theory is a natural next step. Can one write a book that they’d be able to read?”

“I think reading isn’t the issue. I’ve read Weil’s Basic Number Theory. At the time I even thought I understood most of it. It’s just that very little stuck in the mind.”

“But readability is the minimal expectation. We want students to be able to read and learn from the book, and, as you point out, to remember it afterward. Can it be done?”

“Well, you’re asking for an existence proof, so the easiest way to give it is to point to an actual example. So here’s this book by Nancy Childress, Class Field Theory. It’s short, and there are lots of exercises. There’s a section called ‘Some General Questions Motivating Class Field Theory.’ The author claims she has used it successfully in a course.”

“Define ‘successfully’!”

“Don’t be snarky. Let’s look at the book.”

“Well, the first thing I notice is that the chapter on Local Class Field Theory is the last one. So she’s not taking the local-global approach. I guess that means she’s going for the classical approach.”

“But she does bring in the idèles in chapter 4, so it’s not 100% classical.”

“Well, of course, you’d need to in order to take care of infinite extensions. Another classical thing about the book, though, is that it uses the analytic point of view quite a bit. Chapter 2 gives Dirichlet’s theorem on primes in arithmetic progressions, introducing L-functions so that they can be used later. Several of the crucial proofs are analytic, which is the way it was done originally.”

“That may well be a good choice. Chevalley’s algebraic purism was not really a good idea, and Weil was probably right when he said that introducing cohomology to do CFT is overkill.”

“But you know, the real problem is this: I read all the proofs, I understand them, and I still don’t have a ‘big picture’ of what has been achieved. She says in the preface that CFT is a ‘strikingly beautiful topic,’ but I still don’t see why.”

“Well, you just read the book, you didn’t take the class. Maybe it takes a semester of thinking about it.”

“On the other hand, maybe that’s the problem with books on CFT. Proving the theorems is hard and requires methods that may well be irrelevant to understanding the theorems. The core ideas are lost in the technicalities.”

“That’s certainly what happened to me when I tried to read Artin-Tate!”

“It happens less here than it does in that book, of course; Artin-Tate was never intended as a textbook. It was really aimed at folks who already knew the theory and might appreciate a new take on it. Plus, it assumed you already knew group cohomology quite well.”

“In Childress’s book the issue is not pre-requisites, though. It’s quite restrained from that point of view, except maybe in the last chapter. It’s just that the proofs are intricate and hard to absorb, and the results are intricate in a way that is orthogonal to the proofs.”

“So is this book an existence proof, or not?”

“Well, I think it proves that one can write up the proofs accessibly, and even get some motivation in. But I’m still looking for a book that will explain the core ideas to me without bogging down on technique. And maybe also demonstrate how to use them.”

“So this is half the job done?”

“Yeah, that’s what I’d say. If you’re going to teach a course in Class Field Theory, I think this is the book you want to use. But you’d still need to do a lot of work to get across to the students what exactly the results say, why they’re so important, and how they can be used.”

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He’s not sure who all these people are nor why they are discussing Class Field Theory, but he does agree with some of what they say. 


Preface.- A Brief Review.- Dirichlet's Theorem on Primes in Arithmetic Progressions.- Ray Class Groups.- The Idelic Theory.- Artin Reciprocity.- The Existence Theorem, Consequences and Applications.- Local Class Field Theory.- Bibliography.- Index.