A Dedekind criterion over valued fields

Let $(K,\nu)$ be an arbitrary-rank valued field, let $R_\nu$ be the valuation ring of $(K,\nu)$, and let $K(\alpha)/K$ be a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give some necessary and sufficient conditions for $R_\nu[\alpha]$ to be integrally closed. We further characterize the integral closedness of $R_\nu[\alpha]$ which is based on information about the valuations on $K(\alpha)$ extending $\nu$. Our results enhance and generalize some existing results as well as provide applications and examples.