# Suggestions for good books on class field theory

Recently I tried to learn class field theory, but I find it is difficult. I have read the book "Algebraic Number Theory" by J. W. S. Cassels and A. Frohlich. In the book, the approach to class field theory is cohomology of groups. Although I have learned cohomology of groups, I find that those theorems in the book are complicated and can not form a system. I'm wondering what are people's opinions of the book above, can you give me some suggestions on learning class field theory, and could you recommend some good books on class field theory?

• Have you tried Milne's notes? May 15, 2010 at 10:19
• Yes, CFT is hard to learn (and hard to teach). The book by Cassels and Frohlich is excellent, but some of the material is rather concisely presented for someone who is learning the material for the first time. (Most other books take more pages to cover less ground.) To students learning CFT for the first time, I recommend Prof. Milne's lecture notes on the subject, available at jmilne.org. May 15, 2010 at 10:23
• For local class field theory, look at Iwasawa's beautiful book "Local Class Field Theory". It is out of print, so find it in a library. You should have a vague understanding of the use of complex multiplication to generate abelian extensions of imaginary quadratic fields first, in order to have the motivation for what is in Iwasawa's book. Aug 19, 2010 at 3:23
• Sorry, does anyone have a link for Wyman's article "What is a Reciprocity Law?" mentioned above? Can't find the complete article in the net. Thank you. Sep 18, 2018 at 18:51

When you are first learning class field theory, it helps to start by getting some idea of what the fuss is about. I am not sure if you have already gotten past this stage, but if not, I recommend B. F. Wyman's article "What is a Reciprocity Law?" in the American Mathematical Monthly, Vol. 79, No. 6 (Jun. - Jul., 1972), pp. 571-586. I also highly recommend David Cox's book Primes of the Form $x^2 + ny^2$ (mentioned by Daniel Larsson). Cox's book will show you what class field theory is good for and will get you to the statements of the main theorems quickly in a very accessible way. (You can safely skim through most the earlier sections of the book if your goal is to get to the class field theory section quickly.) As a bonus, the book will also give you an introduction to complex multiplication on elliptic curves.

However, Cox's book does not prove the main theorems of class field theory. You will need to look elsewhere for the proofs. There are several different approaches and someone else's favorite book may be unappealing to you and vice versa. You will have to dip into several different books and see which approach appeals to you. One book that has not been mentioned yet is Serge Lang's Algebraic Number Theory. Even if you ultimately choose not to use Lang's book as your main text, there is a short essay by Lang in that book, summarizing the different approaches to class field theory, that is worth its weight in gold.

The obvious answers for beginners, bound to come up at some point, are:

• Nancy Childress' recent book "Class field theory", Springer

• David Cox's "Primes on the form $x^2+ny^2$", Wiley (I think)

Besides these the notes by Milne mentioned in the comments above are really excellent, as is the approach by Neukirch (as is given in his book "Algebraic Number Theory") even though this is rather abstract.

• I am not familiar with Childress' book, having completed the long, pain-staking process of finding a CFT literature collection that satisfies me before it came out. Can you -- or someone else -- speak to the merits of Childress' book? Why would one look there instead of Milne's notes, for instance? May 15, 2010 at 18:05
• I would rate Childress's notes as slightly more accessible for beginners than Milne's notes. May 15, 2010 at 18:55
• @Pete: Hmmm, the merits... It's been a while since I looked in it (and I don't have at hand now) but the most obvious I can think of now is that is relatively easy to digest right after a course in basic algebraic number theory. No cohomology, no central simple algebras, no abstract class formations etc. The down-side is that it is, in my opinion, not so quick coming to the central ideas and theorems of CFT. But on the other hand this can also be seen as one of its strong sides since it instead travels the classical, chronological route. May 15, 2010 at 19:44
• ...Advising a student I would probably urge her/him to browse through Childress' book getting a feel for the theory, perhaps spending a few hours on it, and then move right up to Milne's notes. Or rather, in fact, first I'd have them browsing Cox's book and then Childress' and Milne's. May 15, 2010 at 19:46

I'm surprised no one has mentioned Algebraic Number Fields by Janusz. This is the most down-to-earth book I know which presents a complete proof of the theorems of global class field theory and which is relatively clear and well-written. The beginning of the book describes the basic theory of algebraic number fields, and the book finishes with class field theory. The proofs use a small amount of group cohomology (you should be fine) and use the original, analytic method to prove the First (or Second depending on the author) Fundamental Inequality.

The formulation of the theorems is the more elementary one using ideal theory (as opposed to Childress, which uses ideles). Once you have learned the ideal-theoretic proofs, you might want to read this article. The idele-theoretic versions of the main theorems may be easily derived from the ideal-theoretic ones.

Unlike the book by Cox mentioned above, Janusz's book contains complete proofs of all the theorems. But, as someone who shares your distaste for Cassels-Frohlich, I think Janusz is fairly easy to follow (that doesn't mean the proofs of class field theory are easy, though!).

You also might find this thread to be of some use.

• Janusz was the first book I tried to learn class field theory from. I found it the clearest for conveying the statements of the theorems, but (even more) difficult than the other usual sources for the proofs. Since I think it is important to have the main result thoroughly in one's head before moving to the proofs, I second the recommendation of Janusz. Aug 19, 2010 at 2:26

Perhaps the article Class field theory summarized (Rocky Mt. J. Math. 11, 195-225 (1981)) by Dennis Garbanati should be contained in such a list as a very accessible introduction. For those who can read German, both Hasse's Marburg lectures and Deuring's notes from Goettingen are excellent. Neukirch's cohomological approach (Klassenkoerpertheorie in regenbogen's list) will be translated into English this year.

Edit: I am a little bit reluctant to put this link here as I can't access this web page anymore; but a couple of links are still working, and google will probably find the rest.

• Since he's apparently too modest to say so himself, I'd like to point out that Dr. Lemmermeyer himself has a very good book on reciprocity laws, which has been a pleasure to dip in now and then. For myself, I've found class field theory persistently difficult and technical over the years. Being provided a mixture of clear mathematics and a broad historical perspective by a knowledgeable author goes a long way towards easing the pain. May 15, 2010 at 19:39
• I completely agree with this! That is an excellent book. My only question is: when does the promised part II arrive? :) May 16, 2010 at 12:31
• The appearance of part 2 was linked to my being in a position to do so. A teaching load of 25 hours per week and a two hours drive to the next university library slow things down somewhat -) I also realized that explaining Kummer and Hecke requires a considerable background in analytic techniques, so my current plans are writing a book on the beginnings of class field theory (Euler, Dirichlet, Kronecker, Kummer) up to the first two inequalities before continuing with part 2. May 16, 2010 at 16:01
• Sounds good! Do let us know when the CLT-book is nearing its completion so we can all be prepared to run to the bookstores :) May 16, 2010 at 16:49

Among the few books on class field theory I tried to read, Weil's Basic Number Theory is the one I found most accessible. By far.

• Maybe it is bacause you are more familiar with Analysis. Feb 9, 2011 at 14:43
• Weil's book determines the kernel of reciprocity map in section 8 of the last chapter, which appears neither in Cassels-Frohlich nor in Milne's notes. Aug 4, 2012 at 21:13

I'm a beginner with basically no background in algebraic number theory, and am close to finishing Number Theory 2: Introduction to Class Field Theory by Kato, Kurokawa, Saito.

I love this book - it does a good job explaining the big pictures of number theory (e.g. the local-global philosophy) and providing motivation. It begins with concrete examples of what class field theory says - this is one thing I wanted. Throughout the book, they state theorems and their consequences and postpone the proofs, but still give proofs eventually. I really like this style, as it prevents getting bogged down in proofs. The last section claims to outline the proof of the main theorems of class field theory, but I haven't read it yet. I have not read Volume 1, and was able to get through Volume 2 without any difficulty.

Algebraic Number Theory by Neukirch is a good one for first learner. I've read Janusz's book, it's also very good. However some treatment in Janusz's book on algebra is not so careful, you could read Zariski's Commutative Algebra to find better explanations.

There is also an older book of Neukirch on Class Field Theory: http://www.mathi.uni-heidelberg.de/~schmidt/Neukirch/index.html

An English translation will appear soon.

Edit: It has appeared: http://www.mathi.uni-heidelberg.de/~schmidt/Neukirch-en/index-de.html

For local class field theory, there is Local Fields and Their Extensions by I. B. Fesenko and S. V. Vostokov: https://www.maths.nottingham.ac.uk/personal/ibf/book/book.html

And "Class Field Theory: From Theory to Practice" by Georges Gras.

Edit (19.09.2018): Here are Uwe Jannsen's lecture notes on class field theory: http://www.mathematik.uni-regensburg.de/Jannsen/Classfieldtheory-gesamt.pdf

Geometric class field theory is easier and covered in Milne's Arithmetic Duality Theorems.

If all you need is the major statements from CFT with a few examples, check out the appendix in Lawrence Washington's "Introduction to Cyclotomic Fields" for a speedy overview of both local and global class field theory. It's not big on exposition, and you won't learn the proofs, but it's short and not time-consuming to read. You can then see some applications of the theory in chapter 11 of the same book.