Criterion for finite Galois extension

Question

I am trying to relax the condition for checking whether a extension is Galois. I got from a textbook that:

$E/F$ is a finite Galois Extension iff $E$ is a spitting field of a separable irreducible polynomial in $F[x]$.

But for some of the example I met, E is not necessarily a spitting field of a separable irreducible polynomial. I am trying to relax the condition:

if $E$ is a splitting field of a polynomial in $F[x]$ whose irreducible factors over $F$ are separable, then $E/F$ is Galois.

I can prove the statement, but I am not sure if my argument is valid. It would be a lot more convenient if it is true.

Here is my proof:

Since $E$ is a splitting filed of a polynomial $f$ in $F[x]$, we have $E/F$ is a normal extension and $E=F[\alpha_1,...,\alpha_n]$ where $\alpha_i$ are roots of $f$. For each $\alpha_i \in E$, $irr(\alpha_i,F)$ is separable by assumption. So, we have $E=F[\alpha_1,...,\alpha_n]$ is separable.

Any suggestion will be appreciated.

Answer 1

Your proof is OK. More generally, you can prove that the compositum of two Galois extensions (inside some fixed algebraic closure) is a Galois extension as well. This also explains the equivalence between these two conditions in a more conceptual way.