Question on separable field extenions

Question

Hi I was given this question which I cannot express myself mathematically on so would indeed like the help and appreciate it I am given $ K/F $ is a finite field extension. I am required to show that K is separable if and only if there is a finite number of separable elements $ \alpha_1, \alpha_2,...,\alpha_n \in K $ such that $ K = F[\alpha_1, \alpha_2,...,\alpha_n] $. First direction is obvious seeing as how if i know it is separable and finite then in particular the finite set of generators is separable but how to show that if K is generated by a finite number of separable elements then K itself is separable? Any help appreciated Thank you all

Answer 1

Let $E$ be the splitting field over of the set of minimal polynomials for $\alpha_1,\dots,\alpha_n$. Since $\alpha_1,\dots,\alpha_n$ are separable, $E/F$ is Galois. Let $\alpha\in K=F[\alpha_1,\dots,\alpha_n]$. Then the minimal polynomial for $\alpha$ over $F$ splits in $E$ (all irreducible polynomials that have a root in a Galois extension, split in that extension). The splitting field $L$ of $\alpha$ is therefore an intermediate field of the Galois extension $E/F$ and therefore $L/F$ is Galois. Every element of a Galois extension of $F$ is separable. Thus $\alpha$ is separable.