Lemma 10.37.5. Any localization of a normal domain is normal.

**Proof.**
Let $R$ be a normal domain, and let $S \subset R$ be a multiplicative subset. Suppose $g$ is an element of the fraction field of $R$ which is integral over $S^{-1}R$. Let $P = x^ d + \sum _{j < d} a_ j x^ j$ be a polynomial with $a_ i \in S^{-1}R$ such that $P(g) = 0$. Choose $s \in S$ such that $sa_ i \in R$ for all $i$. Then $sg$ satisfies the monic polynomial $x^ d + \sum _{j < d} s^{d-j}a_ j x^ j$ which has coefficients $s^{d-j}a_ j$ in $R$. Hence $sg \in R$ because $R$ is normal. Hence $g \in S^{-1}R$.
$\square$

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