Automorphisms and Galois Theory

FrobeniusAutomorphism(L) : RngLocA -> Map

Return the automorphism of the unramified extension L which is the lift of the frobenius automorphism on the residue class field of L.

AutomorphismGroup(L) : RngLocA -> Grp, Map

Return the automorphism group of the local field L and a map from the group to the parent of automorphisms of L.

DecompositionGroup(L) : RngLocA -> GrpPerm

InertiaGroup(L) : RngLocA -> GrpPerm

RamificationGroup(L, i) : RngLocA, RngIntElt -> GrpPerm

Return the subgroup of the automorphism group of the local field L whose elements are the automorphisms (represented as group elements) σsuch that v(σ(z) - z) ≥i + 1. The decomposition group is the -1th ramification group and the inertia group is the 0th ramification group.

FixedField(L, G) : RngLocA, GrpPerm -> RngLocA

Return the subfield of the local field L which is fixed by the automorphisms (represented as group elements) in the subgroup G of the automorphism group of L.

Example RngLocA_auto-gal (H51E6)

> P<x> := PolynomialRing(Integers());
> L := LocalField(pAdicField(7, 50), x^6 - 49*x^2 + 686);
> A, am := AutomorphismGroup(L);
> am(Random(A));
Mapping from: RngLocA: L to RngLocA: L
> $1(L.1);
-(279674609046925265141076018485*7^-2 + O(7^34))*$.1^5 + O(7^35)*$.1^4 + 
    (1035905251748988129458881464123*7^-1 + O(7^35))*$.1^3 + O(7^36)*$.1^2 - 
    (1009443907710864908501983735501 + O(7^36))*$.1 + O(7^37)
> FixedField(L, A);
Extension of 7-adic field mod 7^50 by (1 + O(7^33))*x + O(7^33)
> InertiaGroup(L);
Permutation group acting on a set of cardinality 6
    Id($)
    (1, 2)(3, 5)(4, 6)
> FixedField(L, InertiaGroup(L));
Extension of 7-adic field mod 7^50 by (1 + O(7^37))*x^3 - (2*7^2 + O(7^37))*x^2 
    + (7^4 + O(7^37))*x - 4*7^6 + O(7^37)