## Tag `09E2`

Chapter 10: Commutative Algebra > Section 10.156: Nagata rings

Lemma 10.156.18. 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 10.156.15) and hence analytically unramified by Lemma 10.156.13. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 43578–43583 (see updates for more information).

```
\begin{lemma}
\label{lemma-nagata-pth-roots}
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 $A$.
\end{lemma}
\begin{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 \ref{proposition-nagata-universally-japanese})
and hence analytically unramified by
Lemma \ref{lemma-local-nagata-domain-analytically-unramified}.
\end{proof}
```

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