Chebotarev density theorem

In mathematics, specifically in algebraic number theory, the Chebotarev density theorem, named after Nikolai Chebotarev, statistically describes the splitting of primes in a given Galois extension K {\displaystyle K} of the field Q {\displaystyle \mathbb {Q} } of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K {\displaystyle K} . There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime p {\displaystyle p} in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p {\displaystyle p} less than a large integer N {\displaystyle N} , tends to a certain limit as N {\displaystyle N} goes to infinity. It was proved by Chebotarev in his thesis in 1922. A special case that is easier to state says that if K {\displaystyle K} is an algebraic number field which is a Galois extension of Q {\displaystyle \mathbb {Q} } of degree n {\displaystyle n} , then the prime numbers that completely split in K {\displaystyle K} have density 1 / n {\displaystyle 1/n} among all primes. More generally, splitting behavior can be specified by assigning to (almost) every prime number an invariant, its Frobenius element, which is a representative of a well-defined conjugacy class in the Galois group Gal ⁡ ( K / Q ) {\displaystyle \operatorname {Gal} (K/\mathbb {Q} )} . In this case, the theorem says that the asymptotic distribution of these invariants is uniform over the group, so that a conjugacy class with k {\displaystyle k} elements occurs with frequency asymptotic to k / n {\displaystyle k/n} .