Chebotarev's density theorem

The tschebotarjowsche leak rate ( depending on the transcription and tightness set of Chebotarev or Tschebotareff ) is a generalization of the theorem of Dirichlet on primes in arithmetic progressions on Galois extensions of number fields. In the case of an abelian extension of one obtains back the proposition that the set of primes of the form, natural resistance really. In its general form, it follows in particular that exactly the primes completely disassembled in a Galois extension of of degree.

The set was found by Nikolai Grigoryevich Chebotaryov in 1922 and 1923 for the first time published in 1925 in Russian, German.

Formulation

Let be a Galois extension of number fields with, and a conjugacy class. Then, the amount of the unbranched prime ideals of whose Frobenius element (in the case of a non- abelian extension in general this is a conjugacy class ) is the same, natural tightness

Applications

For an abelian extension, for example, in quadratic fields, each conjugacy class has exactly one element, which is why we obtain a uniform distribution. If the non- abelian group of order, so the conjugacy classes consist of 1, 3 and 2 elements such that the prime ideals of fully decomposed into three prime ideals in exactly two split ( with inertia degree and ) and are sluggish.

One can also conclude that it is a composite number is exactly one irreducible polynomial over a global body, so over all local completions is reducible it. For example, this applies to each with Galois group isomorphic to the group of four small between.

You can also get information about the structure of the Galois group and this narrow probabilistically with the tschebotarjowschen tightness theorem on the decomposition of a polynomial in residual bodies.

Decomposes modulo almost all primes completely into linear factors, it also disintegrates over completely; this is a kind of local-global principle. If an irreducible polynomial with integer coefficients, the modulo almost all primes has a zero, so it has degree.

Are Galois extensions of a number field, and is the set of prime ideals of which are broken down into or fully, but finitely many exceptions equal, it follows. ( In this case, the condition that the extensions are Galois, are not dropped. ) A Galois extension is thus uniquely determined by the set of prime ideals fully disassembled. So in order to classify the Galois extensions of, it is sufficient to determine the quantities of prime ideals of which may occur as sets of prime ideals fully disassembled. This happens for abelian extensions just by the class field theory; for non- abelian extensions, this is still an unsolved problem, see the Langlands program.