On a Theorem of Dedekind

Let $(K,ν)$ be an arbitrary valued field with valuation ring $R_ν$ and $L=K(α)$, where $α$ is a root of a monic irreducible polynomial $f\in R_ν[x]$. In this paper, we characterize the integral closedness of $R_ν[α]$ in such a way that extend Dedekind's criterion. Without the assumption of separability of the extension $L/K$, we show that Dedekind's theorem and its converse hold.

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