unramified action

Let K be a number fieldMathworldPlanetmath and let ν be a discrete valuationPlanetmathPlanetmath on K (this might be, for example, the valuationMathworldPlanetmath attached to a prime idealMathworldPlanetmathPlanetmathPlanetmath 𝔓 of K).

Let Kν be the completion of K at ν, and let 𝒪ν be the ring of integersMathworldPlanetmath of Kν, i.e.


The maximal idealMathworldPlanetmath of 𝒪ν will be denoted by


and we denote by kν the residue fieldMathworldPlanetmath of Kν, which is


We will consider three different global Galois groupsMathworldPlanetmath, namely


where K¯,Kν¯,kν¯ are algebraic closuresMathworldPlanetmath of the corresponding field. We also define notation for the inertia group of GKν¯/Kν

Definition 1.

Let S be a set and suppose there is a group actionMathworldPlanetmath of Gal(Kν¯/Kν) on S. We say that S is unramified at ν, or the action of GKν¯/Kν on S is unramified at ν, if the action of Iν on S is trivial, i.e.


Remark: By Galois theoryMathworldPlanetmath we know that, Kνnr, the fixed field of Iν, the inertia subgroupMathworldPlanetmathPlanetmath, is the maximal unramified extensionPlanetmathPlanetmath of Kν, so

Title unramified action
Canonical name UnramifiedAction
Date of creation 2013-03-22 13:56:26
Last modified on 2013-03-22 13:56:26
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Definition
Classification msc 11S15
Synonym set is unramified at a valuation
Related topic InfiniteGaloisTheory
Related topic DecompositionGroup
Related topic Valuation