The automorphisms of a local ring or field are determined by their images on the inertial element generating the inertia ring and on the uniformizing element. All computations necessary to determine the automorphism group can be performed in the local ring.
Let L be a local ring with uniformizing element pi and denote a root of unity of order p^f - 1 (not necessarily the inertial element) by a. Every automorphism of L is determined by the images of pi and a. Since the inertia ring is isomorphic to a Galois extension of the p-adic ring by a, its automorphisms are induced by the powers of the Frobenius automorphism of GF(p, f), mapping a to a^p. Hence, the possible images of a are a^((p^i)) with 0 <= i < f. For each of these possible images the roots of the image of the Eisenstein polynomial of L are determined, thus giving the images of pi.This function returns the automorphisms of L as a sequence of maps of L into L.
Return the automorphism group of the local ring L as a permutation group (representing the regular action). Also return the map from the permutation group to the group of automorphisms represented explicitly (i.e. like returned from the function above).
> I<p,a> := LocalRing(2, 2, 10);
> R<x> := PolynomialRing(I);
> L<b> := LocalRing(I, x^2 + 2*a*x + 2*a^2);
> L;
Local Ring with Eisenstein polynomial x^2 + 2*a*x - 2*a - 2 over I
> A := Automorphisms(L);
> [<A[i](L.1), A[i](L.2)> : i in [1 .. #A]];
[
<b + O(b^18), a>,
<-2*a - b + O(b^18), a>,
<a*b + O(b^18), -a - 1>,
<2*a + 2 - a*b + O(b^18), -a - 1>
]
> AutomorphismGroup(L);
Permutation group G acting on a set of cardinality 4
Order = 4 = 2^2
(1, 4)(2, 3)
(1, 2)(3, 4)
Mapping from: GrpPerm: $, Degree 4, Order 2^2 to Power Structure of Map given by
a rule [no inverse]
We see that L is a Galois extension with Galois group V_4.