A Universe of Sorts

Recall that an extension is galois if it is separable and normal.
Consider some separable extension $L/K$ .
By primitive element, can be written as $L = K(\alpha)$
Since $L$ is separable, the minimal polynomial of $\alpha$ , $p(x) \in K[x]$ is separable, and so splits into linear factors.
Build the splitting field $M$ of $p(x)$ . This will contain $L$ , as $L = K(\alpha) \subseteq K(\alpha, \beta, \gamma, \dots)$ where $\alpha, \beta, \gamma, \dots$ are the roots of $p(x)$ .
This is normal (since it is the splitting field of a polynomial).
This is separable, since it is generated by separable elements $\alpha$ , $\beta$ , $\gamma$ , and so on.

§ Separable Extension Is Contained in Galois Extension