# Learning roadmap for Class Field Theory and more

I consider to study Class Field Theory and Advance Number Theory by my self this next semester. However, there are many books, lecture notes... I would like to choose the most comprehensive book to work on. In my case, I have solid back ground in Abstract Algebra, and I also have knowledge in Commutative Algebra and Homological Algebra.

Please feel freely suggest me some books to work on. If one never mind, please also write down the roadmap for student in the first year who want to study and work on Number Theory in the future.

Thanks for reading!

• j.s. milne course notes online (for algebraic number theory and class field theory), lang "algebraic number theory" (hard to read sometimes, for me at least), neukirch "algebraic number theory" – yoyo Dec 13 '11 at 0:24
• There are MathOverflow threads (one, two) that you should read. – Dylan Moreland Dec 13 '11 at 4:44
• I'm giving this as a comment because I never read the book: what about the book of Kato, Kurokowa and Saito (the second part of a trilogy, I read the first part). I think it will at least motivate you but maybe for a better treatment you will need more references. – quantum Sep 17 '18 at 12:44

## 2 Answers

Cassels and Fröhlich is still the best reference for the basics of Class Field Theory, in my view. Cox's book, recommended by lhf, is also a good place to get motivation, historical and cultural background, and an overview of the theory.

Also the article What is a reciprocity law by Wyman is a helpful guide.

The key point to grasp is that there are two a priori quite distinct notions: class fields, which are Galois extensions of number fields characterized by the fact that primes in the ground field split in the extension provided they admit generators satisfying certain congruence conditions (e.g. the extension $\mathbb Q(\zeta_n)$ of $\mathbb Q$, in which a prime $p$ splits completely if and only if it is $\equiv 1 \bmod n$); and abelian extensions, i.e. Galois extensions of number fields with abelian Galois group (e.g. the extension $\mathbb Q(\zeta_n)$ of $\mathbb Q$, whose Galois group over $\mathbb Q$ is isomorphic to $(\mathbb Z/n)^{\times}$).

The main result of class field theory is that these two classes of extensions coincide (as the example of $\mathbb Q(\zeta_n)$ over $\mathbb Q$ illustrates). This fundamental fact can get a bit lost in the discussion of the Artin map, idèles, Galois cohomology, and so on, and so it is good to keep it in mind from the beginning, and to consider all the material that you learn in the light of this fact.

As for a more general road-map, that is a bit much for one question, but you could look at this guide on MO to learning Galois representations.

• That was helpful. Thanks you very much! – Knumber10 Dec 13 '11 at 4:25
• Dear MattE, suppose I would like to study the Fermat's last theorem(or now Fermat-Wiles-Taylor theorem), which subjects(except Algebraic Geometry) should I study and what are good book on these subject? – Knumber10 Dec 13 '11 at 5:02
• @NguyễnDuyKhánh Dear Nguyen, For FLT and related topics, you should get the book "Modular forms and Fermat's Last Theorem", edited by Cornell, Silverman, and Stevens. This is the basic graduate student reference. You will need more than CFT to understand it. Many of the references in the Galois representations thread on MO linked to in my answer will be relevant, and of course Silverman's book on elliptic curves is another basic prerequisite. But I would start with the Cornell--Silverman--Stevens book, and work backwards from that to more basic things that you need to know. Regards, – Matt E Dec 13 '11 at 5:10
• @NguyễnDuyKhánh It's also worth adding that the main (the only?) reason why some people are afraid of Cassels-Fröhlich is the somewhat dry unmotivated introduction of group cohomlogy before anything else. Since you say that you have a solid background in commutative/homological algebra, that book will be perfect for you, since the actual number theoretic chapters by Serre and Tate are just wonderful. – Alex B. Dec 13 '11 at 6:57

See Primes of the Form $x^2+ny^2$, by David Cox. See also Best book ever on Number Theory.