Corollary 3

— *Let $G$ be the group of $K$-automorphisms of $\Omega$. The set of invariants of $G$ in $\Omega$ is the relative p-radical closure of $K$ in $\Omega$ ($V$, p. 25).

In other words, an element $x$ of $\Omega$ is p-radical over $K$ if and only if it has no conjugate other than itself.

Let $p$ be the characteristic exponent and let $x \in \Omega$. By $V$, p. 44, Prop. 13, there exists an integer $m \geq 0$ such that $y = x^{p^m}$ is algebraic and separable over $K$. Now $x$ is invariant under $G$ if and only if $y$ is invariant under $G$; by Cor. 2 this is the case if and only if $y$ is of degree 1 over $K$, which amounts to saying that $y \in K$. Now the corollary is an immediate consequence.*