The field automorphisms of the cyclotomic field (see ChapterΒ 18) are given by the linear maps on that are defined by E
E
, where and Gcd
hold (seeΒ GaloisCyc
(18.5-1)). Note that this action is not equal to exponentiation of cyclotomics, i.e., for general cyclotomics , is different from .
(In GAP, the image of a cyclotomic under can be computed as GaloisCyc(
)
.)
gap> ( E(5) + E(5)^4 )^2; GaloisCyc( E(5) + E(5)^4, 2 ); -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4 E(5)^2+E(5)^3
For Gcd
, the map E
E
does not define a field automorphism of but only a -linear map.
gap> GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 ); 2 -6