Given a number field K, a Galois extension field L, and prime ideals p of K and P of L unramified over p, there exists a unique element σ = ((L/K), P) of the Galois group G = Gal(L/K) such that for every element α of L, σ(α)≅α^(N(p)) (mod P), where N(p) is the norm of the prime ideal p in K. The symbol ((L/K), P) is called an Artin symbol. If L is an Abelian extension of K, the Artin symbol ((L/K), P) depends only on the prime ideal p of K lying under P, so it may be written as ((L/K), p). In this case, the Artin symbol can be generalized as follows. Let a be an ideal of K with prime factorization a = product_(i = 1)^r p_i^(e_i).

Artin map | class field | class field theory | Hilbert class field