Ask Your Question

Revision history [back]

click to hide/show revision 1
initial version

The easiest way is probably to write the linear factor that the norm form splits off over the number field and compute its norm (which is the norm form) as a resultant taken with the minimal polynomial of the field generator:

sage: M.<a>=NumberField(x^3-x-8)
sage: B=[M(a) for a in M.maximal_order().basis()]
sage: R.<x0,x1,x2,a>=QQ[]
sage: f=sum([R.gen(i)*B[i].lift()(a) for i in [0,1,2]])
sage: f.resultant(a^3-a-8,a)
x0^3 + x0^2*x1 - 6*x0*x1^2 + 8*x1^3 + 2*x0^2*x2 - 11*x0*x1*x2 + 44*x1^2*x2 + x0*x2^2 + 92*x1*x2^2 + 64*x2^3

You should generally try to avoid working over splitting fields. It's rarely necessary and, for larger degree extensions, often infeasible.