constructive method of proving algebraic closure?

May 2010
Hi everyone, i'm new. Nice to be here.
I'm not a native english speaker, so please forgive my far-from-perfect use of this language.
My question is:

Let alpha and beta be two algebraic numbers, and let P and Q be the respective minimal polinomials. So P(alpha)=0 and Q(beta)=0.
Is there a way to express, in terms of P and Q, a polinomial R such that R(alpha+beta)=0 holds?
Similarly, is there a way to find S such that S(alpha*beta)=0 holds?

Such polinomials R and S must exist, because the set of algebraic nubers is a field, but i don't know if there is a way to find some kind of function (or at least an algorithm) f such that f(P,Q)=R. Hope this is clear.

Thank you for your attention.

Edit: the title is not correct. Solving this problem has little to do with proving algebraic closure, i think. Sorry guys.
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