Thaine’s theorem

Let F/ be a totally real abelian number field. By the Kronecker-Weber theoremMathworldPlanetmath, there exists an m such that F(ζm). Let G be the Galois group of the extensionPlanetmathPlanetmathPlanetmath F/Q. Let 𝒪F× denote the group of units in the ring of integersMathworldPlanetmath of F, let C be the subgroup of 𝒪F× consisting of units η of the form


for some collectionMathworldPlanetmath of ba. (Here, N denotes the norm operator and ζm is a primitive m-th root of unityMathworldPlanetmath.) Finally, let A denote the ideal class groupPlanetmathPlanetmathPlanetmath of F.

Theorem 1 (Thaine).

Suppose p is a rational prime not dividing the degree [F:Q] and suppose θZ[G] annihilates the Sylow p-subgroup of E/C. Then 2θ annihilates the Sylow p-subgroup of A.

This is one of the most sophisticated results concerning the annihilatorsMathworldPlanetmathPlanetmathPlanetmath of an ideal class group. It is a direct, but more complicated, version of Stickelberger’s theorem, applied to totally real fields (for which Stickelberger’s theorem gives no information).

Title Thaine’s theorem
Canonical name ThainesTheorem
Date of creation 2013-03-22 14:12:34
Last modified on 2013-03-22 14:12:34
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Theorem
Classification msc 11R29