tower of separable extension

Question

Let $K⊂L⊂M$ be a tower of fields. Let $L/K$ and $M/L$ be separable, is it true that $M/K$ is separable?

I guess there are counterexamples, but I cannot point out them. Thank you for your kind help.[I know if we substitute separable to normal, there are easy example.]

Answer 1

Yes, it is! Let us first show that it is true for finite extensions and then reduce to the finite case later:

Assume that $M/L$ and $L/K$ are finite separable extensions. Recall that a finite extension is separable if and only if its degree of separability is equal to the degree of the extension. This leaves us with: $$[M:K]_s=[M:L]_s\cdot [L:K]_s=[M:L]\cdot [L:K]=[M:K]$$ So we are done with the finite case.

Let's assume that $M/L$ and $L/K$ are (not necessarily finite) separable extensions. Let $\alpha\in M$. Assume that $a_0,...,a_n\in L$ are the coefficients of the minimal polynomial of $\alpha$ over $L$. We know that the extension $K(\alpha,a_0,\dots,a_n)/K$ is finite. Define $$F:=K(\alpha,a_0,\dots,a_n)\cap L$$ Since $F\subset L$, we know that $F/K$ is separable. Also the minimal polynomial of $\alpha$ over $F$ is the same as that over $L$ and hence separable. Thus the extension $K(\alpha,a_0,\dots,a_n)/K$ is seperable by the first paragraph since all extensions are finite.

Answer 2

By the primitive element theorem, $L=K(\alpha),M=L(\beta)=K(\alpha)(\beta)$ for some $\alpha\in L,\beta \in M$. Then the minimal polynomial $m_{\beta, L} \mid m_{\beta, K}$, whence $m_{\beta, L}$ is separable. Suppose $m_{\beta, K}$ is inseparable, i.e. the characteristic of $K$ is $p\gt 0$ and $m_{\beta, K}=f(x^p)$ for some irreducible polynomial $f(x)$. Then $m_{\beta, L}=g(x^p)$ for some irreducible polynomial $g(x)$ and the characteristic of $L=K(\alpha)$ is also p, which implies that $m_{\beta, L}$ is inseparable, a contradiction. So $\beta$ is separable over $K$ and thus $M/K$ is separable.