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Let $R$ be an integral domain with algebraically closed fraction field. Let $P$ be a prime ideal of $R[Y_1, \ldots, Y_r]$. Then it follows that $R[Y_1, \ldots, Y_r]_{P}$ is an integral domain. I would like to prove that $R[Y_1, \ldots, Y_r]_{P}$ is integrally closed in its fraction field. Any explanation would be appreciated. Thank you.

Edit. I am asking this question because I wanted to prove that $\mathbb{P}^r_R$ is a normal scheme when $R$ is a valuation ring with an algebraically closed fraction field.

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    $\begingroup$ If $R$ is a valuation ring then it is integrally closed, then so is any polynomial ring over $R$, and its localizations as well. $\endgroup$
    – user26857
    Mar 24, 2020 at 19:40
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    $\begingroup$ @user26857 that looks like an answer to me - would you care to record it as such? $\endgroup$
    – KReiser
    Mar 24, 2020 at 19:48

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This is a community wiki answer recording the discussion from the comments, in order that this question might be marked as answered (once this post is upvoted or accepted).

If $R$ is a valuation ring then it is integrally closed, then so is any polynomial ring over $R$, and its localizations as well. – user26857

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