Valuation rings are integrally closed (${\overline{V}}^{\mathrm{Frac}V}=V$)

If an element $x$ of the fraction field is absent from a valuation ring $V$, its inverse $y$ belongs to the maximal ideal. A monic equation for $x$, multiplied by a power of $y$, would put $1$ in that maximal ideal. Thus $V$ is an integrally closed domain, without a discrete or Noetherian hypothesis.