Definition 2

Let E be an extension of a field K. We shall say that E is p-radical $^1$ (over K) if every element of E is p-radical over K. When this is so, E is said to be of finite height when the set of heights of elements of E is bounded above and we call the height of E the maximum of the heights of its elements.

We note that every p-radical extension of a perfect field (in particular of a field of characteristic 0) is trivial.

Let K be a field.. The p-radical extensions of height 0 of K are the trivial extensions. Every p-radical extension of K is algebraic. If E is a p-radical extension of K and F a p-radical extension of E, then F is a p-radical extension of K : for, given any $x \in F$, there exists an integer $m \geq 0$ such that $x^{p^m} \in E$ and an integer $n \geq 0$ such that $(x^{p^m})^{p^n} \in K$, that is, $x^{p^m + n} \in K$.