ray class field

Proposition 1.

Let L/K be a finite abelian extensionMathworldPlanetmath of number fieldsMathworldPlanetmath, and let OK be the ring of integersMathworldPlanetmath of K. There exists an integral ideal COK, divisible by precisely the prime idealsPlanetmathPlanetmath of K that ramify in L, such that


where ((α),L/K) is the Artin mapMathworldPlanetmath.

Definition 1.

The conductorPlanetmathPlanetmathPlanetmath of a finite abelian extension L/K is the largest ideal CL/KOK satisfying the above properties.

Note that there is a “largest ideal” with this condition because if propositionPlanetmathPlanetmathPlanetmath 1 is true for 𝒞1,𝒞2 then it is also true for 𝒞1+𝒞2.

Definition 2.

Let I be an integral ideal of K. A ray class field of K (modulo I) is a finite abelian extension KI/K with the property that for any other finite abelian extension L/K with conductor CL/K,


Note: It can be proved that there is a unique ray class field with a given conductor. In words, the ray class field is the biggest abelian extension of K with a given conductor (although the conductor of K does not necessarily equal !, see example 2).

Remark: Let 𝔭 be a prime of K unramified in L, and let 𝔓 be a prime above 𝔭. Then (𝔭,L/K)=1 if and only if the extensionPlanetmathPlanetmathPlanetmath of residue fields is of degree 1


if and only if 𝔭 splits completely in L. Thus we obtain a characterizationMathworldPlanetmath of the ray class field of conductor 𝒞 as the abelian extension of K such that a prime of K splits completely if and only if it is of the form



  1. 1.

    The ray class field of of conductor N is the Nth-cyclotomic extension of . More concretely, let ζN be a primitive Nth root of unityMathworldPlanetmath. Then

  2. 2.

    so the conductor of (i)(2)/ is (1).

  3. 3.

    K(1), the ray class field of conductor (1), is the maximal abelian extension of K which is unramified everywhere. It is, in fact, the Hilbert class fieldMathworldPlanetmath of K.


Title ray class field
Canonical name RayClassField
Date of creation 2013-03-22 13:54:01
Last modified on 2013-03-22 13:54:01
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Definition
Classification msc 11R37
Synonym conductor
Related topic ArtinMap
Related topic ExistenceOfHilbertClassField
Related topic NumberField
Related topic AnExactSequenceForRayClassGroups
Defines conductor of an extension