Lemma 10.37.14. Let $R$ be a normal ring. Then $R[x]$ is a normal ring.

**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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