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Given the ring $R = \mathbb{Q}[[x_1, \dots, x_n]]$ and a prime ideal $\mathfrak{p}$ of $R$, I would like to be able to say whether or not $R / \mathfrak{p}$ is integrally closed. In particular, I'd like to know what conditions on $\mathfrak{p}$ guarantee that $R/\mathfrak{p}$ is an integrally closed domain.

More generally, is it known when a quotient of a regular local ring is an integrally closed domain?

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    $\begingroup$ You can use Serre criterion of normality: $R/\mathfrak{p}$ is integrally closed if and only if it is regular in codimension 1 and has property $S_2$. The first condition can be checked using the Jacobian criterion, the second is more delicate but holds for instance if $\mathfrak{p}$ is generated by a regular sequence. $\endgroup$
    – abx
    Jan 23, 2018 at 6:04
  • $\begingroup$ Thanks! I knew Serre's conditions were necessary and sufficient but didn't know any simple ways to check for them in $R/\mathfrak{p}$. This helps. $\endgroup$
    – aleph_aleph_null
    Jan 24, 2018 at 3:17

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