About minimal polynomial of an $p$th root (Field extension whose characteristic is $p$)

Question

I want to know if there is a counterexample: Let $k\subset K$ be a field extension whose characteristic is $p$, where $p$ is a prime. For an $a\notin k$, suppose that $a^p\in k$. Then $x^p-a^p$ is the minimal polynomial for $a$ over $k$.

Answer 1

There is no counterexample, since this is actually true. Obviously $a\in K$ is a root of $f=X^p-a^p\in k[x]$. Now, $f$ factors as $(X-a)^p$ over $\overline{k}$. Since $a\not\in k$, we conclude by the following well-known lemma (try to prove it yourself, otherwise you can find a proof on MSE) that $f$ is the minimal polynomial of $a$ over $k$.

Lemma. Let $K$ be a field (not necessarily of characteristic $p>0$) and $a\not\in K^p$, then $f=X^p-a\in K[X]$ is irreducible if and only if $f$ has no roots in $K$.