TITLE: (Simple?) applications of Class Field Theory?
QUESTION [12 upvotes]: Does anyone know any simple/nice applications of class field theory?
I would really like to find one related to diophantine equations, but anything you got would be good.
Thanks

REPLY [5 votes]: If you have studied global class field theory then there are many uses. Firstly the Cebotarev density theorem appears as a generalisation of Dirichlet's theorem...and it is a nice little exercise to see how Dirichlet's theorem falls out if you explicitly use the cyclotomic extension $\mathbb{Q}(\zeta_n)/\mathbb{Q}$.
Also the Artin reciprocity law explains all previously known reciprocity laws and provides a more general setting for them...it gives a reciprocity law for every single abelian extension! For example, quadratic reciprocity follows from it. Read Cox to find out how.
Cox also is a master of the subject of primes of the form $x^2 + ny^2$. Suffice to say it is easy to see that this is really the same question as determining which primes split into nice ideals in the order $\mathbb{Z}[\sqrt{-n}]$ of $\mathbb{Q}(\sqrt{-n})$. Depending on what $n$ is you can find major uses for the Hilbert class fields and the ring class fields in getting congruence conditions for such a prime to have the above form! Which path you use depends on whether $\mathbb{Z}[\sqrt{-n}]$ is the full ring of integers.