6.4 Fields over Subfields

F / G

The quotient operator / evaluates to a new field H. This field has the same elements as F, i.e., is a domain equal to F. However H is viewed as a field over the field G, which must be a subfield of F.

What subfield a field is viewed over determines its Galois group. As described in GaloisGroup the Galois group is the group of field automorphisms that leave the subfield fixed. It also influences the results of Norm, Trace, CharPol, and MinPol, because they are defined in terms of the Galois group.

    gap> F := GF(2^12);
    GF(2^12)
    gap> G := GF(2^2);
    GF(2^2)
    gap> Q := F / G;
    GF(2^12)/GF(2^2)
    gap> Norm( F, Z(2^6) );
    Z(2)^0
    gap> Norm( Q, Z(2^6) );
    Z(2^2)^2 

The operator / calls G.operations./( F, G ).

The default function called this way is FieldOps./, which simply makes a copy of F and enters G into the record component F.field (see Field Records).

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