Chebotarev's density theorem
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In mathematics, Chebotarev's density theorem in algebraic number theory is a generalisation to algebraic number fields that are Galois extensions, of Dirichlet's theorem on arithmetic progressions. It reduces to that theorem (in the form of a density statement) in the case of an abelian extension of the rational number field. In its general form, it in particular implies that only 1/n of prime numbers split completely into degree 1 prime ideals, in a Galois extension of Q of degree n.
The statement of the theorem is in terms of the Frobenius element of a prime (ideal), which is in fact an associated conjugacy class C of elements of the Galois group G. If we fix C then the theorem says that asymptotically a proportion
- |C|/|G|
of primes have associated Frobenius element as C. When G is abelian the classes of course each have size 1. For the case of a non-abelian group of order 6 they have size 1, 2 and 3, and there are correspondingly (for example) 50% of primes p that split into two prime ideals exactly, in an extension of Q with it as Galois group.
The result is due to Nikolai Grigoryevich Chebotaryov, in 1922. (Chebotaryov is literally transliterated as Chebotarëv, where the ë is actually pronounced yo; the same as for Khrushchev, in fact.)
