60.2-5 GaloisStabilizer
‣ GaloisStabilizer( F )( attribute )

Let F be an abelian number field (see IsAbelianNumberField (60.2-3)) with conductor n 𝑛 n , say. (This means that the n 𝑛 n -th cyclotomic field is the smallest cyclotomic field containing F, see Conductor (18.1-7).) GaloisStabilizer returns the set of all those integers k 𝑘 k in the range [ 1 . . n ] fragments [ 1 . . n ] [1..n] such that the field automorphism induced by raising n 𝑛 n -th roots of unity to the k 𝑘 k -th power acts trivially on F.

gap> r5:= Sqrt(5);
E(5)-E(5)^2-E(5)^3+E(5)^4
gap> GaloisCyc( r5, 4 ) = r5;  GaloisCyc( r5, 2 ) = r5;
true
false
gap> GaloisStabilizer( Field( [ r5 ] ) );
[ 1, 4 ]