Valuation rings are integrally closed
= Valuation rings are integrally closed
{title2=$\overline V^{\operatorname{Frac}V}=V$}
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.