Show that an extension is separable

Question

Let $K$ be a field with $\operatorname{char} K=p$, where $p$ is a prime, and let the degree of the extension $K \leq L$ be coprime to $p$.

How can I show that the extension is separable??

Could you give me some hints??

Answer 1

If for a field $K$ we have $char(k)=p$ and $f\in K[x]$ is monic and irreducible, then $f$ is separable or there exists a separable polynomial $g\in K[x]$ such that there exists $n\in \mathbb{Z}_{\geq 1}$ with $f(x)=g(x^{p^n})$. Suppose this is the case for the minimal polynomial of some $\alpha\in L$,then de degree of this polynomial is divisible by $p$, so $[L:K]=[L:K(\alpha)][K(\alpha):K]$ is divisible by $p$ which is a contradiction, because you assumed that the degree of the extension was coprime with $p$.

Answer 2

Hint. What can you say about the degree of the minimal polynomial of a non-separable element?