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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.
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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