Let $K/\mathbb{Q}$ be a galois extension of degree $d$, and let $\alpha \in K$. Let's look at $\prod_{\sigma\in Gal(K,\mathbb{Q})}\sigma(\alpha)$. This is a rational number $q\in\mathbb{Q}$. Say we know $H(q)$, that is, if $q=\frac{m}{n}$ in reduced form we know $|m|,|n|\leq H$. Is it possible to bound $H(\alpha)$?
I know that if $f$ is the minimal polynomial then one has $h(\alpha)deg(\alpha)=m(f)$ where $m$ is the mahler measure, but I can't seem to bound in this case the mahler measure of $f$ by $h(f(0))$. Also I tried using the fact that galois conjugates have the same height, but I can't seem to bound the height of a product from below in general.
Any help would be apprecieated! Thanks.