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Concept# Chebotarev's density theorem

Summary

Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field \mathbb{Q} of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime 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 less than a large integer N, tends to a certain limit as N goes to infinity. It was proved by Nikolai Chebotaryov in his thesis in 1922, published in .
A special case that is easier to state says that if K is an algebraic number field which is a Galois extension of \mathbb{Q} of degree n, then the prime numbers that completely split in K have density
:1/n
among all primes.

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