# Criterion for finite Galois extension

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.