Lemma 10.162.18. Let $(A, \mathfrak m)$ be a Noetherian local domain which is Nagata and has fraction field of characteristic $p$. If $a \in A$ has a $p$th root in $A^\wedge$, then $a$ has a $p$th root in $A$.

Proof. Consider the ring extension $A \subset B = A[x]/(x^ p - a)$. If $a$ does not have a $p$th root in $A$, then $B$ is a domain whose completion isn't reduced. This contradicts our earlier results, as $B$ is a Nagata ring (Proposition 10.162.15) and hence analytically unramified by Lemma 10.162.13. $\square$

Comment #2461 by on

The second sentence of the statement has an extra "is": it should probably read "…then $a$ has a $p$th root in $A$." In the proof, the phrase "as $B$ is a Nagata" should probably say "…is Nagata" or "…is a Nagata ring".

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