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 x F, when constructed as an associative algebra. DirectSum(Algebra(F,F), Algebra(F,F));
Given an element g of a permutation group G, satisfying G simeq Aut(F), returns an element α ∈Aut(F) corresponding to g.
Given a function, an intrinsic, or a map between fields a describing an automorphism a : F to F, returns the corresponding field automorphism.
The identity automorphism of the field F.
If K = F x F, return the swap involution.
If F simeq K, return the element of Aut(K) corresponding to a ∈Aut(F).
The field F for which a ∈Aut(F).
The order of a.
The fixed field of a.
The map defined by a : F to F.
Returns -1, 0 or 1 if P is inert, ramified, or split, respectively in the fixed field of a.
Returns true iff a is the identity automorphism.
The automorphism given by a b, namely x |-> a(b(x)).
The automorphism an.
The automorphism a - 1.
Returns true iff a and b are automorphisms of the same field F, and a = b.
Evaluate a(x).
Given a vector v = (x1, ...., xn) in an F-vector space V, apply a to each coordinate, to get a(v) = (a(x1), ..., a(xn)).
Given a matrix M = (xij) with entries in F, apply a to each coordinate, to get a(M) = (a(xij)).
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).
Mult: RngIntElt Default: Order(a)
The image of the map x |-> ∑j=0m - 1 ∑aj(x) on P, where m is determined by Mult.
Mult: RngIntElt Default: Order(a)
The image of the map x |-> ∏j=0m - 1 ∑aj(x) on P, where m is determined by Mult.