Changing Coefficient Ring

The ChangeRing function enables the changing of the coefficient ring of a polynomial ring or ideal.

ChangeRing(I, S) : RngMPol, Rng -> RngMPol
Given an ideal I of a polynomial ring P=R[x1, ..., xn] of rank n with coefficient ring R, together with a ring S, construct the ideal J of the polynomial ring Q=S[x1, ..., xn] obtained by coercing the coefficients of the elements of the basis of I into S. It is necessary that all elements of the old coefficient ring R can be automatically coerced into the new coefficient ring S. If R and S are fields and R is known to be a subfield of S and the current basis of I is a Gröbner basis, then the basis of J is marked automatically to be a Gröbner basis of J.

Example GB_ChangeRing (H112E12)

It is better to find the Gröbner basis of an ideal over the smallest subfield possible (e.g. Q), then use ChangeRing to create the equivalent ideal over a splitting field to find the variety. 
> P<x, y, z, t, u> := PolynomialRing(RationalField(), 5);
> I := ideal<P |
>     x + y + z + t + u,
>     x*y + y*z + z*t + t*u + u*x,
>     x*y*z + y*z*t + z*t*u + t*u*x + u*x*y,
>     x*y*z*t + y*z*t*u + z*t*u*x + t*u*x*y + u*x*y*z,
>     x*y*z*t*u - 1>;
> Groebner(I);
> K<W> := CyclotomicField(5);
> J := ChangeRing(I, K);
> V := Variety(J);
> #V;
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V2.28, 13 July 2023