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!
Best Answer
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.