Editing Chebotarev density theorem (section)

{{short description|Describes statistically the splitting of primes in a given Galois extension of Q}}
The '''Chebotarev density theorem''' in [[algebraic number theory]] describes statistically the splitting of [[prime number|primes]] in a given [[Galois extension]] ''K'' of the field <math>\mathbb{Q}</math> of [[rational number]]s. Generally speaking, a prime integer will factor into several [[Ideal number|ideal primes]] in the ring of [[algebraic integer]]s 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 {{harv|Tschebotareff|1926}}.

A special case that is easier to state says that if ''K'' is an [[algebraic number field]] which is a Galois extension of <math>\mathbb{Q}</math> of degree ''n'', then the prime numbers that completely split in ''K'' have density 

: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'').

Then the theorem says that the asymptotic distribution of these invariants is uniform over the group, so that a conjugacy class with ''k'' elements occurs with frequency asymptotic to 

:''k''/''n''.