16.8 GaloisGroup for Extension Fields

GaloisGroup( L )

returns the Galois group of the field L if L is a normal extension and issues an error if not. The Galois group is a group of extension automorphisms (see ExtensionAutomorphism).

The computation of a Galois group is computationally relatively hard, and can take significant time.

    gap> g:=GaloisGroup(f);
    Group( ExtensionAutomorphism(AlgebraicExtension(GF(2),Z(2)^0*(y^
    2 + y + 1)),RootOf(Z(2)^0*(y^2 + y + 1))+Z(2)^0) )
    gap> h:=GaloisGroup(e);
    Group( ExtensionAutomorphism(e,alpha^3+
    3*alpha), ExtensionAutomorphism(e,-1*alpha), ExtensionAutomorphism(e,
    -1*alpha^3-3*alpha) )
    gap> Size(h);
    4
    gap> AbelianInvariants(h);
    [ 2, 2 ]

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