Proof. Let $\mathfrak q$ be a prime of $R[x]$. Set $\mathfrak p = R \cap \mathfrak q$. Then we see that $R_{\mathfrak p}[x]$ is a normal domain by Lemma 10.37.8. Hence $(R[x])_{\mathfrak q}$ is a normal domain by Lemma 10.37.5. $\square$

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