absolute values of algebraic numbers under Galois automorphism This could be very easy question, but I have no idea about it in depth.

Q. Let $\alpha$ be an algebraic integer. Let $\sigma$ be an automorphism of Galois group of $\mathbb{Q}(\alpha)$ (over $\mathbb{Q}$). If, as complex number, $|\alpha|<1$, is it necessary that $|\sigma(\alpha)|<1$? 


In general, no.

For example, let $\alpha = 2-\sqrt{2},\,$ and consider the automorphism $\sigma$ of $\mathbb{Q}(a)$ such that $\sigma(\sqrt{2}) = -\sqrt{2}$.

Then 
$$|a| = |2-\sqrt{2}| < 1$$
but
$$|\sigma(a)| = |2+\sqrt{2}| > 1$$
