Degree of non-separable extension

Question

Suppose $K$ is a field with charasteristic number $p$. Further suppose that L is a non-separable extension of K with $[L:K]=k$. Why does it hold that k is a multiple of p?

Answer 1

If you take for granted the non-trivial fact that the set of elements in $L$ that are separable over $K$ forms a subfield $K^s$, then this can be proved as follows. Choose $x\in L$ that is not separable over $K$. It is elementary that, for some $n\geq 1$, $x^{p^n}\in K^s$ (necessarily $n\geq 1$ since $x$ is not separable). This means that $x$ is purely inseparable over $K^s$, i.e., its minimal polynomial over $K^s$ is of the form $x^{p^m}-a$ for some $m\geq 1$ and some $a\in K^s$ ($m\geq 1$ because $x\notin K^s$). The extension $K^s(x)$ is then of degree $p^m$, and since $K\subseteq K^s\subseteq K^s(x)\subseteq L$, $[L:K]$ is divisible by $[K^s(x):K^s]=p^m$.