Automorphisms of Fields

Magma now (from V2.29) contains a new structure for field automorphisms. A field automorphism is of type FldAut. Also supports the swap automorphism of \(F \times F\), when constructed as an associative algebra. DirectSum(Algebra(F,F), Algebra(F,F));

Creation of Field Automorphisms

FieldAutomorphism(F, g): Fld, GrpPermElt -> FldAut
FieldAutomorphism(F, g): AlgAss[Fld], GrpPermElt -> FldAut

Given an element \(g\) of a permutation group \(G\), satisfying \(G \simeq {\operatorname{Aut}}(F)\), returns an element \(\alpha \in {\operatorname{Aut}}(F)\) corresponding to \(g\).

FieldAutomorphism(F, a): Fld, Map[Fld,Fld] -> FldAut
FieldAutomorphism(F, a): Fld, Intrinsic -> FldAut
FieldAutomorphism(F, a): Fld, UserProgram -> FldAut
FieldAutomorphism(F, a): AlgAss[Fld], Map[AlgAss[Fld],AlgAss[Fld]] -> FldAut

Given a function, an intrinsic, or a map between fields \(a\) describing an automorphism \(a : F \to F\), returns the corresponding field automorphism.

IdentityAutomorphism(F): Fld -> FldAut

The identity automorphism of the field \(F\).

FieldAutomorphism(K): AlgAss[Fld] -> FldAut

If \(K = F \times F\), return the swap involution.

ChangeRing(a, K): FldAut, Fld -> FldAut
ChangeRing(a, K): FldAut, AlgAss[Fld] -> FldAut

If \(F \simeq K\), return the element of \({\operatorname{Aut}}(K)\) corresponding to \(a \in {\operatorname{Aut}}(F)\).

Properties of Field Automorphisms

BaseField(a): FldAut -> Fld

The field \(F\) for which \(a \in {\operatorname{Aut}}(F)\).

Order(a): FldAut -> RngIntElt

The order of \(a\).

FixedField(a): FldAut -> Fld

The fixed field of \(a\).

Automorphism(a): FldAut -> Map[Fld, Fld]

The map defined by \(a : F \to F\).

FixedFieldSymbol(a, P): FldAut, RngOrdIdl -> RngIntElt
FixedFieldSymbol(a, P): FldAut, RngInt -> RngIntElt

Returns \(-1,0\) or \(1\) if \(P\) is inert, ramified, or split, respectively in the fixed field of \(a\).

Predicates on Field Automorphisms

IsIdentity(a): FldAut -> BoolElt

Returns true iff \(a\) is the identity automorphism.

Arithmetic of Field Automorphisms

a * b: FldAut, FldAut -> FldAut

The automorphism given by \(a \circ b\), namely \(x \mapsto a(b(x))\).

a ^ n: FldAut, RngIntElt -> FldAut

The automorphism \(a^n\).

Inverse(a): FldAut -> FldAut

The automorphism \(a^{-1}\).

a eq b: FldAut, FldAut -> BoolElt

Returns true iff \(a\) and \(b\) are automorphisms of the same field \(F\), and \(a = b\).

x @ a: FldElt, FldAut -> FldElt
x @ a: RngOrdElt, FldAut -> RngOrdElt
x @ a: RngIntElt, FldAut -> RngIntElt
x @ a: AlgAssElt[Fld], FldAut -> AlgAssElt[Fld]

Evaluate \(a(x)\).

v @ a: ModTupFldElt[Fld], FldAut -> ModTupFldElt[Fld]

Given a vector \(v = (x_1, ...., x_n)\) in an \(F\)-vector space \(V\), apply \(a\) to each coordinate, to get \(a(v) = (a(x_1), \ldots, a(x_n))\).

M @ a: ModMatFldElt[Fld], FldAut -> ModMatFldElt[Fld]
M @ a: AlgMatElt[Fld], FldAut -> AlgMatElt[Fld]
M @ a: GrpMatElt[Fld], FldAut -> GrpMatElt[Fld]
M @ a: AlgMatElt[AlgAss[Fld]], FldAut -> AlgMatElt[AlgAss[Fld]]

Given a matrix \(M = (x_{ij})\) with entries in \(F\), apply \(a\) to each coordinate, to get \(a(M) = (a(x_{ij}))\).

I @ a: RngOrdFracIdl[FldOrd], FldAut -> RngOrdFracIdl[FldOrd]
I @ a: RngInt, FldAut -> RngInt
I @ a: RngIntFracIdl, FldAut -> RngIntFracIdl

Given a fraction ideal \(I\) of an order \(R\), and an automorphism \(a\) of the fraction field of \(R\), return the fractional ideal \(a(I)\).

Trace(P, a): RngOrdFracIdl, FldAut -> RngOrdFracIdl
Trace(P, a): RngIntFracIdl, FldAut -> RngIntFracIdl
Mult: RngIntElt                    Default: Order(a)

The image of the map \(x \mapsto \sum_{j=0}^{m-1} \sum a^j(x)\) on \(P\), where \(m\) is determined by Mult.

Norm(P, a): RngOrdFracIdl, FldAut -> RngOrdFracIdl
Norm(P, a): RngIntFracIdl, FldAut -> RngIntFracIdl
Mult: RngIntElt                    Default: Order(a)

The image of the map \(x \mapsto \prod_{j=0}^{m-1} \sum a^j(x)\) on \(P\), where \(m\) is determined by Mult.

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