  Let $(K,\nu)$ be an arbitrary valued field with valuation ring $R_{\nu}$ and
$L=K(\alpha)$, where $\alpha$ is a root of a monic irreducible polynomial $f\in
R_{\nu}[x]$. In this paper, we characterize the integral closedness of
$R_\nu[\alpha]$ 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.
