Prime Degree Field Extension Separable

Question

Let $p$ be a prime and $L$ is a prime degree field extension of $K$, therefore $[K: L]=p$. Why is then $L$ separable over $K$?

Answer 1

From Serge Lang, Algebra, 2002, p.254, exercise 14:

Let $\text{char}\; K = p.$ Let $L$ be a finite extension of $K$, and suppose $[L : K]$ prime to $p$. Show that $L$ is separable over $K$.

Maybe you meant this? In which case, it follows from the fact that the minimal polynomial of any purely inseparable element $\alpha\in L$ over $K$ has degree $p^{\mu}$ for some integer $\mu$, and hence $p^{\mu}\mid [L:F]$, which can only be the case when $\mu=0$ i.e. $L$ is separable over $K$.