upper bound for discriminant of a number field

Question

Assume that $f$ is an irreducible polynomial of degree $d$ with Galois group $S_d$, fix a root $\theta\in \mathbb C$ with $f(\theta)=0$ and define the number field extension $K=\mathbb Q(\theta)$. Can we give an upper bound for the absolute value of the discriminant in terms of the coefficients of $f$ and $d$? I am guessing that something like $(100 d |f|)^{100 d} $ might be possible? Here $|f|$ is the maximum absolute value of the coefficients of $f$.