Lemma 15.115.8. Let $A$ be a discrete valuation ring. Assume the reside field $\kappa _ A$ has characteristic $p > 0$ and that $a \in A$ is an element whose residue class in $\kappa _ A$ is not a $p$th power. Then $a$ is not a $p$th power in $K$ and the integral closure of $A$ in $K[a^{1/p}]$ is the ring $A[a^{1/p}]$ which is a discrete valuation ring weakly unramified over $A$.

Proof. This lemma proves itself. $\square$

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