Computing minimal polynomials using Groebner bases and the elimination theorem.
alpha := sqrt(2)+sqrt(3);
F := [z-s-t,s^2-2,t^2-3];
G := Groebner[Basis](F,plex(s,t,z));
m := G[1];
factor(m);
eval(m,z=alpha);
expand(eval(m,z=alpha));
G := Groebner[Basis](F,lexdeg([s,t],[z]));
solve(G[2],t);
Groebner[UnivariatePolynomial](z,F);
r1 := resultant(z-s-t,s^2-2,s);
r2 := resultant(r1,t^2-3,t);