Suggestions for good books on class field theory - MathOverflow

Suggestions for good books on class field theory

2010-05-15T10:17:50Z

Recently I learn class field theory, but I find it is difficulty. 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 learnd 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 learnning class field theory, and could you recommend some good books on class field theory?

Answer by Daniel Larsson for Suggestions for good books on class field theory

2010-05-15T10:33:05Z

The obvoius 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.

Answer by Pierre-Yves Gaillard for Suggestions for good books on class field theory

2010-05-15T11:04:16Z

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.

Answer by Franz Lemmermeyer for Suggestions for good books on class field theory

2010-05-15T19:00:03Z

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.

Answer by Timothy Chow for Suggestions for good books on class field theory

2010-05-16T02:27:15Z

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.

Answer by David Corwin for Suggestions for good books on class field theory

2010-08-19T00:17:37Z

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.

Answer by Hong Liu for Suggestions for good books on class field theory

2010-08-19T02:11:45Z

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.

Answer by Timo Keller for Suggestions for good books on class field theory

2011-12-10T21:11:49Z

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.

Answer by Jeff H for Suggestions for good books on class field theory

2011-12-11T04:38:12Z

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.