Szego Seminar: "Chebotarëv's Density Theorem"

Speaker: John Naughton, Washington University in Saint Louis

Abstract: Dirichlet's Theorem on primes in arithmetic progressions tells us that prime numbers are equally likely to fall into any arithmetic progression which "makes sense" (i.e. those for which the common difference and the initial value are coprime). Meanwhile, Frobenius' Theorem says that the manner in which a polynomial with integer coefficients factors modulo various primes is related to the Galois group of the splitting field of that polynomial. At first, these theorems may seem only vaguely related. They both involve prime numbers - as do many things in life worth thinking about - and they each ask how certain objects are distributed across possible classes. As it happens the two results are quite closely related, and by carefully choosing a polynomial, Frobenius' Theorem almost gives us Dirichlet's. In this talk, we will attempt to reconcile their differences and construct a version of Frobenius' theorem from which Dirichlet's theorem falls out for free. This strengthening of Frobenius' Theorem leads us to discover our titular hero: Chebotarev's density theorem. 

Host: Jeremy Cummings