Stickelberger’s theorem

Theorem 1 (Stickelberger).

Let L=Q(ζm) be a cyclotomic fieldMathworldPlanetmath extensionPlanetmathPlanetmathPlanetmath of Q with Galois group G={σa}a(Z/mZ)×, and consider the group ring Q[G]. Define the Stickelberger element θQ[G] by


and take βZ[G] such that βθZ[G] as well. Then βθ is an annihilatorMathworldPlanetmathPlanetmathPlanetmath for the ideal class groupPlanetmathPlanetmathPlanetmath of Q(ζm).

Note that θ itself need not be an annihilator, just that any multipleMathworldPlanetmathPlanetmath of it in [G] is.

This result allows for the most basic between the (otherwise hard to determine) of a cyclotomic ideal class group and its of annihilators. For an application of Stickelberger’s theorem, see Herbrand’s theorem.

Title Stickelberger’s theorem
Canonical name StickelbergersTheorem
Date of creation 2013-03-22 14:12:31
Last modified on 2013-03-22 14:12:31
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Theorem
Classification msc 11R29
Defines Stickelberger element