16.10 Field functions for Algebraic Extensions

As already mentioned, algebraic extensions are fields. Thus all field functions like Norm and Trace are applicable.

    gap> Trace(r^4+2*r);
    14
    gap> Norm(ran);
    305

DefaultField always returns the algebraic extension, which contains the Algebraic Extension Elements.

    gap> DefaultField(r^2);
    e

As subfields are not yet supported, Field will issue an error, if several elements are given, or if the element is not a primitive element for its default field.

You can create a polynomial ring over an algebraic extension to which all functions described in Ring Functions for Polynomial Rings can be applied, for example you can factor polynomials. Factorization is done --- depending on the polynomial --- by factoring the squarefree norem or using a hensel lift (with possibly added lattice reduction) as described in Abb89, using bounds from BTW93.

    gap> X(e).name:="X";;
    gap> p1:=EmbeddedPolynomial(PolynomialRing(e),p1);
    X^2 + 3*X + 1
    gap> Factors(p1);
    [ X + (-1*alpha^2), X + (alpha^2+3) ]

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