I'm asked to find a finite separable field extension of a non-perfect field $K$ of characteristic $p > 0$ that has degree divisible by $p$, but I don't see the solution. Since $K$ is not perfect I should use an element $\alpha$ that can't be written as $k^p$ for some $k\in K$ and $p=char(K)$. Its minimal polynomial is however clearly not separable. I've found that an Artin-Schreier polynomial $X^n-x-\beta$ for some $\beta \neq \gamma^n -\gamma, \forall \gamma \in K$ satisfies the requirements, I however don't see how to prove the existence of such a $\beta$.
EDIT: Maybe something like this works but it doesn't feel quite right. Let $\alpha$ be an element that cannot be written as $k^p$ for $k\in K$ and $p=char(K)$. Then the polynomial $f=X^p - \alpha$ is irreducible in $K$. Let $\beta$ be a root of $f$ in its splitting field over $K$. Then $L:=K(\beta, \beta^2, ..., \beta^p)=K(\beta)(\beta^2)...(\beta^n)$ is separable since $f$ contains at most $deg(f)=p$ roots, and $[L:K]=p^n$ for some $n\leq p$, hence $p$ divides $[L:K]$
EDIT2: Never mind, this makes no sense whatsoever...