GAP Manual: 6.12. Conjugates

Conjugates( z )
Conjugates( F, z )

In the first form Conjugates returns the list of conjugates of the field element z in its default field over its prime field (see DefaultField). In the second form Conjugates returns the list of conjugates of the field element z in the field F over the subfield F.field. In either case the list may contain duplicates if z lies in a proper subfield of its default field, respectively of F.

The conjugates of an element z in a field F over a subfield S are the roots in F of the characteristic polynomial of z in F (see CharPol). If F is a normal extension of S, then the conjugates of z are the images of z under all elements of the Galois group of F over S (see GaloisGroup), i.e., under those automorphisms of F that leave S fixed. The number of different conjugates of z is given by the degree of the smallest extension of S in which z lies.
For a normal extension F, Norm (see Norm) computes the product, Trace (see Trace) the sum of all conjugates. CharPol (see CharPol) computes the polynomial that has precisely the conjugates with their corresponding multiplicities as roots, MinPol (see MinPol) the squarefree polynomial that has precisely the conjugates as roots.

    gap> Conjugates( Z(2^6) );
    [ Z(2^6), Z(2^6)^2, Z(2^6)^4, Z(2^6)^8, Z(2^6)^16, Z(2^6)^32 ]
    gap> Conjugates( GF(2^12), Z(2^6) );
    [ Z(2^6), Z(2^6)^2, Z(2^6)^4, Z(2^6)^8, Z(2^6)^16, Z(2^6)^32, Z(2^6),
      Z(2^6)^2, Z(2^6)^4, Z(2^6)^8, Z(2^6)^16, Z(2^6)^32 ]
    gap> Conjugates( GF(2^12)/GF(2^2), Z(2^6) );
    [ Z(2^6), Z(2^6)^4, Z(2^6)^16, Z(2^6), Z(2^6)^4, Z(2^6)^16 ] 

The default function FieldOps.Conjugates applies the automorphisms of the Galois group of F (see GaloisGroup) to z and returns the list of images. For nonabelian extensions, this is overlayed by a factorization of the characteristic polynomial.