Finite Separable Extensions

Question

In trying to prove the primitive element theorem. But first how do I prove that if $E/F$ is a finite separable extension of $F$, then there is an extension $K$ such that $K/F$ is Galois? Since E is a separable extension of $F$ should I find a normal extension for $E$?

Answer 1

As $E/F$ is finite, we have that $E=F[\alpha_1,\ldots,\alpha_n]$. Let $f_i$ be the minimal polynomial for $\alpha_i$, then each $f_i$ is a separable polynomial, because $E/F$ is a separable extension. Hence the polynomial $\prod_i f_i$ is a separable polynomial. Take $K$ to be the splitting field for $\prod_i f_i$. $K$ is a splitting field, hence is Galois, and it contains $E$.