Lemma 10.37.8. Let $R$ be a normal domain. Then $R[x]$ is a normal domain.

Proof. The result is true if $R$ is a field $K$ because $K[x]$ is a euclidean domain and hence a principal ideal domain and hence normal by Lemma 10.37.6. Let $g$ be an element of the fraction field of $R[x]$ which is integral over $R[x]$. Because $g$ is integral over $K[x]$ where $K$ is the fraction field of $R$ we may write $g = \alpha _ d x^ d + \alpha _{d-1}x^{d-1} + \ldots + \alpha _0$ with $\alpha _ i \in K$. By Lemma 10.37.7 the elements $\alpha _ i$ are integral over $R$ and hence are in $R$. $\square$

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