Dirichlet's Theorem on primes in arithmetic progression states that for any coprime numbers and , we can find infinitely many primes congruent to modulo . A stronger version is in fact true: each residue class contains the same amount of prime numbers: if you write a table of all prime numbers up to and count how many end up in a given residue class (ie how many end up being congruent to modulo ), you will find that there are approximately . This means primes are equidistributed amongst the residue classes, as there are precisely integers between and that are coprime to , and hence only possible infinite families.

Something somewhat similar happens when looking at the splitting of primes in the ring of Gaussian integers : the behaviour is essentially described by Fermat's Theorem on primes expressible as the sum of two squares. Indeed, any prime congruent to mod factors as the product of two prime elements of (for example, ) whereas primes congruent to mod stay prime in this larger ring (lastly, ramifies in this ring: , which means that gains a repeated factor). So here we see that on average, half of the primes split into two factors, and half stay prime: this is related to the fact that the degree , as a Galois field extension, is 2.

This last observation actually generalises much further, and in fact this generalisation was formulated by Frobenius, who was unable to prove it. The Russian mathematician Chebotarev, however, was able to come up with a proof; this generalisation also gives us Dirichlet's Theorem as a special case. The method of proof, that Chebotarev allegedly thought of while carrying water from the lower part of his town to the higher part, has been invaluable in the development of class field theory. Indeed, when Artin first formulated his acclaimed general reciprocity law, he was unable to prove it; only later, after reading Chebotarev's proof, was he able to come up with a proof of his theorem.

Come to MS.05 to learn about all this, and more! After which we factor into the pub.