When I was considering extensions of non-Archimedean valuation $|\cdot| : K \rightarrow \mathbb{R}$, I came across the problem :
If $L/K$ is an algebraic field extension and $R$ is a valuation ring of $K$, then for $\alpha \in L$, does $N_{K(\alpha)/K}(\alpha) \in R$ imply that $\alpha$ is integral over $R$ ?