Lemma 1

Let K be a field and a an element of K such that $a \notin K^p$. Then for every integer $e \geq 0$, the polynomial $f(X) = X^{p^e} - a$ is irreducible in $K[X]$.

Let $\Omega$ be an algebraic closure of $K$ and let b be the element $a^{p^{-e}}$ of $\Omega$; we denote by g the minimal polynomial of b over $K$. We have $f(X) = (X - b)^{p^e}$ and hence every irreducible polynomial in $K[X]$ which divides $f$ admits b as root, and hence is equal to g. There exists thus (IV, p. 13, Prop. 13) an integer $q \geq 1$ such that $f = g^q$; since q divides the degree $p^e$ of $f$, there exists an integer $e'$ such that $0 \leq e' \leq e$ and $q = p^{e'}$. If c is the constant term of g, we have $-a = c^q$; since we assumed that a does not lie in $K^p$, we must have $q = 1$, that is, $f = g$, so the lemma follows.