# 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) $\sigma$ such that $v(\sigma(z) - z) \ge i + 1$. The decomposition group is the $-1$th ramification group and the inertia group is the $0$th 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: Auto Gal (ex-180d7f)`

The automorphism and inertia groups of a local field are computed and their fixed fields examined.

```magma
> 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)

```