Assume $K$ is a finite extension of $Q$, and $|\cdot|_v$ is an absolute value. Some questions still puzzle me.
(1) If $|\cdot|v$ is non-archimedean, then we know that $|\mathbb Z|_v\leq 1$. Let $y\in \mathcal O_K$, how to prove that $|y|_v\leq 1$?
(2) For any absolute value $|\cdot|_v$, and any $x\in K$, can we prove that all Galois conjugations of $x$ contained in $K$ must have the same absolute value as $x$? Can you prove it without using further knowledge of places but only using basic Galois theory?
Thank you!