Splitting field of a set of separable polynomials implies separability of extension.

Question

Let $F$ be a splitting field of $S\subset K[x]$ over $K$, where $S$ is a set of separable polynomials. I want to show that $F$ is separable over $K$, meaning for all $u\in F-K$, the irreducible polynomial of $u$ in $K$ is separable.

A long way to see this is to use the fact that $F$ is algebraic and Galois over $K$, but I want give a direct proof without talking about Galois.

Answer 1

Look at the Old Answer by Pierre-Yves Gallaird to Splitting field of a separable polynomial is separable. He argues that the following three statements are equivalent for a field extension $L/K$ of degree $d < \infty$:

1. $L/K$ is generated by separable elements;
2. Every element of $L$ is separable over $K$;
3. There are exactly $d$ embeddings of $L$ in $\overline L = \overline K$ over $K$.

For your case, you'd additionally have to argue that an element $u \in F$ is already contained in the splitting field $L \subseteq F$ of a finite subset of $S$.