The following is a well-known result:
Theorem: If and are integers, with , then there are infinitely many primes congruent to .
It turns out that Dirichlet’s Theorem is actually a special case of Artin’s Reciprocity Law. So, we’ll discuss how this works.
Let be an extension of number fields. (That is, and are finite extensions of .) Let and be the rings of integers of and , respectively. (This means that and are the integral closures of in and , respectively.) Now, let be a nonzero prime ideal in . Then for some primes of and some positive integers . If , we say that is ramified over . We call the ramification index. The primes are said to lie above .
Since and are Dedekind domains, and are maximal ideals. Hence and are finite fields, and is a field extension. Let . We call the residue degree.
It is not too difficult to show that if , then . If is a Galois extension, then and are independent of (since the Galois group of over acts transitively on the ), so we can write .
Now, let’s define a few subgroups of . We’ll assume from now on that is a Galois extension. Furthermore, we fix a prime of , and some prime of lying above . Now, define . We call the decomposition group. We now have a homomorphism . To define , we note that an element of permutes cosets of and thus gives the desired homomorphism. Furthermore, this homomorphism is surjective. The kernel of is called the inertia group. Hence .
It is not hard to determine the sizes of and in terms of quantities we already understand: and . In particular, if is an unramified prime, then .
That’s particularly nice, because Galois groups of extensions of finite fields are always cyclic, generated by the Frobenius automorphism. Thus in the unramified case, is cyclic and generated by an automorphism satisfying the congruence for all . (We can extend to all of by multiplicativity.) Furthermore, this element is unique. The common notation for is .
If and are two primes lying above , then there is some element so that . It is easy to verify that . Therefore, if is abelian, then depends only on . In this case, we may write for this element.
Finally, we’re ready to state (part of) the Artin reciprocity theorem.
Let be an abelian Galois extension of number fields, and let be fixed. Then there are infinitely many primes of that are unramified and so that .
(In fact, only finitely many primes ramify, since primes ramify if and only if they divide the discriminant, which can be easily verified. The other part of the statement is more interesting.)
Let’s look at one example. Take and , where . Then , where the isomorphism is as follows: if , then there is an automorphism defined by . Now, if , then . In particular, this case of the theorem is equivalent to Dirichlet’s Theorem on primes in arithmetic progressions.
It turns out that the theorem isn’t too much more general than this, since any abelian extension is contained in a cyclotomic extension (this is the Kronecker-Weber Theorem), and it’s not hard to see what happens to Frobenius elements when we pass to sub-extensions.
All this material can be found (with many more details included) in Serre’s Local Fields.