6.7 GaloisGroup

GaloisGroup( F )

GaloisGroup returns the Galois group of the field F as a group (see Groups) of field automorphisms (see Field Homomorphisms).

The Galois group of a field F over a subfield F.field is the group of automorphisms of F that leave the subfield F.field fixed. This group can be interpreted as a permutation group permuting the zeroes of the characteristic polynomial of a primitive element of F. The degree of this group is equal to the number of zeroes, i.e., to the dimension of F as a vector space over the subfield F.field. It operates transitively on those zeroes. The normal divisors of the Galois group correspond to the subfields between F and F.field.

    gap> G := GaloisGroup( GF(4096)/GF(4) );;
    gap> Size( G );
    6
    gap> IsCyclic( G );
    true    # the Galois group of every finite field is
            # generated by the Frobenius automorphism
    gap> H := GaloisGroup( CF(60) );;
    gap> Size( H );
    16
    gap> IsAbelian( H );
    true 

The default function FieldOps.GaloisGroup just raises an error, since there is no general method to compute the Galois group of a field. This default function is overlaid by more specific functions for special types GaloisGroup for Number Fields).

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