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
IdentityMapping( F ).
Previous Up Top Next