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.