§ Separable Extension is contained in Galois extension

• 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.