6.10 Norm

Norm( z )
Norm( F, z )

In the first form Norm returns the norm of the field element z in its default field over its prime field (see DefaultField). In the second form Norm returns the norm of z in the field F over the subfield F.field.

The norm of an element z in a field F over a subfield S is (-1)^{F.degree/S.degree}=(-1)^{[F:S]} times the constant term of the characteristic polynomial of z (see CharPol). Thus the norm lies in S. The norm is the product of all conjugates of z in the normal closure of F over S (see Conjugates).

    gap> Norm( Z(2^6) );
    Z(2)^0
    gap> Norm( GF(2^12), Z(2^6) );
    Z(2)^0
    gap> Norm( GF(2^12)/GF(2^2), Z(2^6) );
    Z(2^2)^2 

The default function FieldOps.Norm multiplies the conjugates of z in F (see Conjugates). For nonabelian extensions, it is overlayed by a function, which obtains the norm from the characteristic polynomial.

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