Algebraic integers with discriminants containing fixed prime divisors

It is proved that any algebraic integer $\alpha$ of degree $n\ge2$ whose discriminant is a product of powers of prescribed primes $p_1,\dots,p_r$ has the form $\alpha=a+\beta p_1^{v_1}\dotsp_r^{v_r}$, where $\alpha,v_1,\dots,v_r$ are rational integers and $\beta$ is an integer whose height does not exceed an effectively defined bound depending $\max(p1,\dots,p_r)$, $r$, and $n$.