unramified action
Let be a number field and let be a discrete valuation
on
(this might be, for example, the valuation
attached to a prime
ideal
of ).
Let be the completion of at , and let
be the ring of integers of , i.e.
The maximal ideal of will be denoted by
and we denote by the residue field of , which
is
We will consider three different global Galois groups, namely
where are algebraic closures of the corresponding field. We also
define notation for the inertia group of
Definition 1.
Let be a set and suppose there is a group action of
on . We say that
is unramified at , or the action of
on is unramified at
, if the action of on is trivial,
i.e.
Remark: By Galois theory we know that,
, the fixed field of , the
inertia subgroup
, is the maximal unramified extension
of
, 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 |