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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