# the algebra of “p-th roots”

Let $R$ be an integral algebra of finite type over a (perfect) field of characteristic $p > 0$. Let $S$ be the integral closure of $R$ in $Q(R)^{1/p}$ where $Q(R)$ is the field of fractions of $R$. Then there is a "$p$-th root" map $\eta: R \to S$, $\eta(r)^p=r$.

Is there a way to desribe $S$ explicitly, without the reference to the fraction field and integral closure? I am not sure that $S$ is the minimal ring that contains all the "$p$-th roots", i.e. an extension of $R$ such that there exists a map $\eta$ from $R$ to it, $\eta(r)^p=r$ for any $r\in R$. If it is not, then I am also interested in explicit description of such ring.

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Are you assuming $R$ is integrally closed? –  jspecter Apr 30 '12 at 14:26
not necessarily, but you may assume it if you want. –  Dima Sustretov Apr 30 '12 at 14:38