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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

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I am told that Cox's _Primes of the form $x^2 + ny^2$_ is a good resource for this. Class field theory can help answer the question of, well, which primes can be represented in the form $x^2 + ny^2$ for fixed $n$. –  Qiaochu Yuan Mar 14 '12 at 22:05
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Cox's book is great, I was wondering if there was any other examples, like x2+ny2, where class field theory helps. –  Matthew Mar 14 '12 at 22:15

4 Answers 4

You might like Class Field Theory and the First Case of Fermat's Last Theorem by Lenstra and Stevenhagen. The first case of FLT is that we cannot have $$x^p+y^p+z^p=0 \ \mathrm{and} \ xyz \not\equiv 0 \mod p.$$

Lenstra and Stevenhagen use computations with $p$-power reciprocity laws in $\mathbb{Q}(\zeta_p)$ to show that, if the first case of FLT were solvable, than $p$ would have many special properties such as: $$2^{p-1} \equiv 1 \mod p^2$$ $$3^{p-1} \equiv 1 \mod p^2$$ $$2p+1, 4p+1, 8p+1, 10p+1 \ \mbox{are all composite}$$

Using these conditions and ones like them, one can rule out all primes under $10^{18}$. These conditions were already known by other methods; the contribution of this article is to provide very short and uniform proofs from Class Field Theory.

This isn't an easy article -- following all of the details involves understanding Class Field Theory very well. But I think it is as well written as possible, given the subject matter.

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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.

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In my very first research paper, I needed extensions of $\mathbb{Q}$ with dihedral Galois group and with lots of ramified primes of a particular kind. Even though these are non-abelian extensions, their existence can be shown using class field theory.

I used this to say something about growth of Selmer groups of elliptic curves, which are Diophantine equations in two variables of degree 3. Not sure if that qualifies for you as applications to Diophantine equations.

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Artin reciprocity:

Let $K$ be an abelian field extension of the number field $F$, then the Dedekind zeta function of $K$ factors into a product of Hecke L functions of $F$ (associated to one dimensional representation of the Galois group).

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Two corrections to a generally good answer: (1) You mean the Dedekind zeta function of $K$, not $F$. (2) The fact that $\zeta_K$ is a product of $L(\chi, s)$ for $\chi$ one dimensional characters of the Galois group is "just" formal manipulation. The fact which uses CFT is that these $L$-series can be written down explicitly using Grossencharacters. –  David Speyer Mar 15 '12 at 14:25
    
Matthew might like to look at my answer math.stackexchange.com/questions/80265/… , where I describe how to write down the splitting of primes in the splitting field of $x^3-x-1$, using that this is a cyclic extension of $\mathbb{Q}(\sqrt{-23})$. –  David Speyer Mar 15 '12 at 14:26

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