Suggestions for good books on class field theory

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'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.