Proposition 2

Let E be an extension of a field K. For every integer $n \geq 0$ let $E_n$ be the set of elements of E which are $p\text{-radical}$ of height $\leq n$ over K, and let $E_r$ be the set of all elements of E that are p-radical over K. Then $(E_n)_{n \geq 0}$ is an ascending sequence of subextensions of E whose union is $E_\infty$, and $E_r$ is the largest p-radical extension of K contained in E.

For each integer $n \geq 0$ the set $E_n$ consists of all elements $x$ of E such that $x^{p^n} \in K$; since the mapping $x \mapsto x^{p^n}$ is an endomorphism of the field E, we deduce that $E_n$ is a subextension of E. The sequence $(E_n)_{n \geq 0}$ is ascending, with union $E_r$ and hence $E_r$ is a subextension of E (V, p. 11, Prop. 3). It is clear that $E_r$ is a p-radical extension of K and that $E_r$ contains every subextension of E which is p-radical over K.