Automorphisms of Local Rings and Fields

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.

Automorphisms(L): RngPad -> [Map]

Automorphisms(L): FldPad -> [Map]

Given a local ring or field \(L\), returns the automorphisms of \(L\) over its \(p\)-adic sub-field \({\mathbb{Q}}_p\) as a sequence of maps of \(L\) into \(L\).

Automorphisms(K, k): FldPad, FldPad -> [Map]

Automorphisms(K, k): RngPad, RngPad -> [Map]

Given a local ring or field \(K\) over \(k\), returns the \(k\)-automorphisms of \(K\) as a sequence of maps of \(K\) into \(K\).

AutomorphismGroup(L): RngPad -> GrpPerm, Map

AutomorphismGroup(L): FldPad -> GrpPerm, Map

Return the automorphism group acting on \(L\) over its \(p\)-adic sub-field \({\mathbb{Q}}_p\) 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).

AutomorphismGroup(K, k): RngPad, RngPad -> GrpPerm, Map

AutomorphismGroup(K, k): FldPad, FldPad -> GrpPerm, Map

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).

IsNormal(K): RngPad -> BoolElt

IsNormal(K): FldPad -> BoolElt

Given a \(p\)-adic ring or field \(K\), test if \(K\) is normal over it’s prime field \({\mathbb{Q}}_p\), ie. if \(K\) admits exactly \(n\) automorphisms where \(n\) is the degree of \(K\).

IsNormal(K, k): RngPad, RngPad -> BoolElt

IsNormal(K, k): FldPad, FldPad -> BoolElt

Given a \(p\)-adic ring or field \(K\), test if \(K\) is normal over the subfield \(k\).

IsAbelian(K, k): FldPad, FldPad -> BoolElt

Given \(p\)-adic fields \(K/k\), test if the automorphism group of \(K\) over \(k\) is abelian.

Continuations(m, L): Map, RngPad -> [Map]

For an automorphism \(m\) of the \(p\)-adic ring \(L\), compute all possible extensions of \(m\) to \(L\).

IsIsomorphic(E, K): RngPad, RngPad -> BooElt

IsIsomorphic(E, K): FldPad, FldPad -> BooElt

For two \(p\)-adic rings or fields, test if they are isomorphic over \({\mathbb{Q}}_p\).

Example: Automorphisms of a Local Ring (ex-b62260)

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We define an extension of \({\mathbb{Z}}_2\) with ramification degree 2 and inertia degree 2 and compute automorphisms.

> 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

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GaloisGroup(f): RngUPolElt[FldPad] -> GrpPerm, SeqEnum, UserProgram

GaloisGroup(f): RngUPolElt[RngPad] -> GrpPerm, SeqEnum, UserProgram

Computes the 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 \(G\to{\operatorname{Aut}}(F/K)\).

Example: Galois Group of a Local Polynomial (ex-286b99)

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> 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)

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