Exercise 32, § 4

Let $A$ be an integral noetherian local ring and $K$ its field of fractions. Assume that $K$ has nonzero characteristic $p$. Assume that $A$ is a Nagata ring, and denote by $\hat{A}$ the completion of $A$. Let $p$ be a minimal prime ideal of $\hat{A}$, and $L$ the field of fractions of $\hat{A}/p$. Let $k$ be a weak field of representatives of $A$, $k'$ the field of representatives of $\hat{A}$ containing the image of $k$ in $\hat{A}$ (p. 78, Exercise 9, (a)). Then $K$ is separable over $k$ if and only if $L$ is separable over $k'$. (Observe that $L$ is separable over $K$ since $A$ is a Nagata ring. If $K$ is separable over $k$, prove that every derivation of $k'$ into $L$ extends to a derivation of $L$ into $L$.)