Artin map

Let L/K be a Galois extensionMathworldPlanetmath of number fieldsMathworldPlanetmath, with rings of integersMathworldPlanetmath 𝒪L and 𝒪K. For any finite prime 𝔓L lying over a prime 𝔭K, let D(𝔓) denote the decomposition groupMathworldPlanetmath of 𝔓, let T(𝔓) denote the inertia group of 𝔓, and let l:=𝒪L/𝔓 and k:=𝒪K/𝔭 be the residue fieldsMathworldPlanetmath. The exact sequencePlanetmathPlanetmathPlanetmathPlanetmath


yields an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath D(𝔓)/T(𝔓)Gal(l/k). In particular, there is a unique element in D(𝔓)/T(𝔓), denoted [L/K,𝔓], which maps to the qth power Frobenius mapPlanetmathPlanetmath FrobqGal(l/k) under this isomorphism (where q is the number of elements in k). The notation [L/K,𝔓] is referred to as the Artin symbolMathworldPlanetmath of the extensionPlanetmathPlanetmathPlanetmathPlanetmath L/K at 𝔓.

If we add the additional assumptionPlanetmathPlanetmath that 𝔭 is unramified, then T(𝔓) is the trivial group, and [L/K,𝔓] in this situation is an element of D(𝔓)Gal(L/K), called the Frobenius automorphismMathworldPlanetmath of 𝔓.

If, furthermore, L/K is an abelian extensionMathworldPlanetmathPlanetmath (that is, Gal(L/K) is an abelian groupMathworldPlanetmath), then [L/K,𝔓]=[L/K,𝔓] for any other prime 𝔓L lying over 𝔭. In this case, the Frobenius automorphism [L/K,𝔓] is denoted (L/K,𝔭); the change in notation from 𝔓 to 𝔭 reflects the fact that the automorphismPlanetmathPlanetmathPlanetmath is determined by 𝔭K independent of which prime 𝔓 of L above it is chosen for use in the above construction.

Definition 1.

Let S be a finite setMathworldPlanetmath of primes of K, containing all the primes that ramify in L. Let IKS denote the subgroupMathworldPlanetmathPlanetmath of the group IK of fractional idealsMathworldPlanetmathPlanetmath of K which is generated by all the primes in K that are not in S. The Artin map


is the map given by ϕL/K(𝔭):=(L/K,𝔭) for all primes 𝔭S, extended linearly to IKS.

Title Artin map
Canonical name ArtinMap
Date of creation 2013-03-22 12:34:55
Last modified on 2013-03-22 12:34:55
Owner djao (24)
Last modified by djao (24)
Numerical id 9
Author djao (24)
Entry type Definition
Classification msc 11R37
Related topic RayClassField
Defines Artin symbol
Defines Frobenius automorphism