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.


1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.