Proposition 1

Let E be an extension of a field K and x a p-radical element of height e over K; put $a = x^{p^e}$. Then $a \in K$ and the minimal polynomial of x over K is $X^{p^e} - a$: Further we have $[K(x):K] = p^e$.

It suffices to prove that the polynomial $X^{p^e} - a$ is irreducible in $K[X]$. By the definition of height of x we have $a \notin K^p$, so that the proposition follows from the lemma: