The automorphisms of a local ring or field are determined by their images on the generators of the ring. All computations necessary to determine the automorphism group can be performed in the local ring.
Given a local ring or field L, returns the automorphisms of L over its p-adic sub-field Qp as a sequence of maps of L into L.
Given a local ring or field K over k, returns the k-automorphisms of K as a sequence of maps of K into K.
Return the automorphism group acting on L over its p-adic sub-field Qp 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).
Return the automorphism group acting on K over its p-adic k 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).
Given a p-adic ring or field K, test if K is normal over it's prime field Qp, ie. if K admits exactly n automorphisms where n is the degree of K.
Given a p-adic ring or field K, test if K is normal over the subfield k.
Given p-adic fields K/k, test if the automorphism group of K over k is abelian.
For an automorphism m of the p-adic ring L, compute all possible extensions of m to L.
For two p-adic rings or fields, test if they are isomorphic over Qp.
> I<a> := ext<pAdicRing(2, 10) | 2>;
> R<x> := PolynomialRing(I);
> L<b> := ext<I | x^2 + 2*a*x + 2*a^2>;
> L;
Totally ramified extension defined by the polynomial
x^2 + (2*a)*x + -2*a - 2 over Unramified extension
defined by the polynomial x^2 + x + 1 over 2-adic ring
mod 2^10
> A := Automorphisms(L);
> [<A[i](a), A[i](b)> : i in [1 .. #A]];
[ <a, b + O(b^18)>, <a, -b + -2*a + O(b^18)>, <-a - 1,
a*b + O(b^18)>, <-a - 1, -a*b + 2*a + 2 + O(b^18)> ]
> AutomorphismGroup(L);
Permutation group acting on a set of cardinality 4
Id($)
(1, 2)(3, 4)
(1, 3)(2, 4)
Mapping from: GrpPerm: $, Degree 4 to Power Structure
of Map given by a rule
Galois group G of a squarefree polynomial f over a p-adic ring or field K. Returns G as a permutation group on the roots of f in its splitting field F, the roots themselves, and a map G to Aut(F/K).
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > G,r,act:=GaloisGroup(x^4-2); > G; // permutation group on roots in r Permutation group G acting on a set of cardinality 4 (1, 4, 2, 3) (1, 2) > GroupName(G); D4 > F<pi>:=Universe(r); F; // splitting field F of f Totally ramified extension defined by the polynomial x^8 + 8*x^7 + 24*x^6 + 32*x^5 + 18*x^4 + 8*x^3 + 12*x^2 + 8*x + 2 over 2-adic field mod 2^20 > sigma:=act(G.1); // an automorphism of F/K > sigma; Mapping from: FldPad: F to FldPad: F given by a rule [no inverse] > sigma(F.1); (43690*pi^7 - 5*pi^6 + 43677*pi^5 - 15*pi^4 + 43687*pi^3 + 43684*pi - 3)*pi + O(pi^136)