18.11 FrobeniusAutomorphism

FrobeniusAutomorphism( F )

FrobeniusAutomorphism returns the Frobenius automorphism of the finite field F as a field homomorphism (see Field Homomorphisms).

The Frobenius automorphism f of a finite field F of characteristic p is the function that takes each element z of F to its p-th power. Each automorphism of F is a power of the Frobenius automorphism. Thus the Frobenius automorphism is a generator for the Galois group of F (and an appropriate power of it is a generator of the Galois group of F over a subfield S) (see GaloisGroup).

    gap> f := GF(16);
    GF(2^4)
    gap> x := FrobeniusAutomorphism( f );
    FrobeniusAutomorphism( GF(2^4) )
    gap> Z(16) ^ x;
    Z(2^4)^2 

The image of an element z under the i-th power of the Frobenius automorphism f of a finite field F of characteristic p is simply computed by computing the p^i-th power of z. The product of the i-th power and the j-th power of f is the k-th power of f, where k is i*j mod (Size(F)-1). The zeroth power of f is printed as IdentityMapping( F ).

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