Chebotarev density theorem

In a finite Galois extension of number fields, unramified primes whose Frobenius elements lie in a given conjugacy class have density equal to that class’s proportion in the Galois group.

Frobenius element: An element of a Galois group describing the action of a prime on residue classes, defined up to conjugacy for an unramified prime. Its conjugacy class is the arithmetic datum whose frequency the theorem predicts.
Splitting of primes in Galois extensions: The factorization of a prime ideal into prime ideals after extending the number field. Frobenius cycle structure determines the residue degrees and number of primes above an unramified prime.
Finite Galois extension: A finite, normal, separable extension of fields. This is the setting that supplies a finite Galois group and Frobenius conjugacy classes.
Nikolai Chebotarev: A Russian mathematician known for proving the density theorem that bears his name. His work established the general density law for Frobenius conjugacy classes.
Hilbert irreducibility theorem: A theorem guaranteeing that many specializations of an irreducible polynomial remain irreducible. Prime-distribution arguments related to Chebotarev help establish specialization results over number fields.
Decomposition group: The subgroup of a Galois group preserving a chosen prime above a given prime of the base field. It contains the Frobenius element and organizes how a prime behaves in the extension.
Splitting completely: The behavior of a prime that factors into distinct degree-one primes in a number-field extension. The identity conjugacy class gives density one divided by the Galois group’s order.
Residue field: The quotient of a local ring by its maximal ideal, capturing arithmetic modulo a prime. Frobenius acts on the residue field by raising elements to the residue-field size.
Felix W. K. Frobenius: A German mathematician whose work shaped group theory and the theory of finite fields. The prime-associated automorphism used in the theorem carries his name.
Inverse Galois problem: The problem of realizing every finite group as a Galois group over a specified field. Chebotarev describes prime behavior once such a Galois realization is known, though it does not solve realization itself.

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