Higher derivations and field extensions

Let $K$ be a field having prime characteristic $p$. The following conditions on a subfield $k$ of $K$ are equivalent: (i) ${ \cap _n}{K^{{p^n}}}(k) = k$ and $K/k$ is separable. (ii) $k$ is the field of constants of an infinite higher derivation defined in $K$. (iii) $k$ is the field of constants of a set of infinite higher derivations defined in $K$. If $K/k$ is separably generated and $k$ is algebraically closed in $K$, then $k$ is the field of constants of an infinite higher derivation in $K$. If $K/k$ is finitely generated then $k$ is the field of constants of an infinite higher derivation in $K$ if and only if $K/k$ is regular.