Consider a polynomial in one variable $x$ with irrational coefficients which are algebraic, i.e., they have a defining polynomial. As an example, take $p(x) = (x-3)(x-\sqrt{2}) = x^2-(3+\sqrt{2})x+3\sqrt{x}$. In my setting, $\sqrt{2}$ could also be some real root $r$ not being a radical expression, but I always have a defining polynomial $q(y)$ for $r$ so that $q(r)=0$.
My question is, whether there is a polynomial $\tilde p(x)$ with rational coefficients such that $p(a) = 0 \Rightarrow \tilde p(a) = 0$ (so I only need an over-approximation of the roots of $p$).
As a side information, such polynomials occur if I start with polynomials in multiple variables and substitute variables by real roots found by another procedure.