TITLE: Are there (finite) field extensions which aren't Galois?
QUESTION [5 upvotes]: Let $K=F(\alpha)$. It strikes me that there are two possibilities:

$\alpha \in F$, in which case $K\cong F$
$\alpha\not\in F$, in which case $K$ is the splitting field of $(x-\alpha)$ (and hence Galois).

(I actually think that even #1 might be described as a Galois extension - $|G(F/F)|=[F:F]$, for example.)
In any case, what does it mean to say that an extension is not Galois?

REPLY [11 votes]: I think the key point you're missing is that the label "Galois" isn't something that applies to the extension field -- it is something that applies to the extension.
It doesn't make sense to say that $K$ is Galois -- instead, what you observed in your question is that the extension $K/K$ is a Galois one, because $K$ is the splitting field of a polynomial with coefficients in $K$.
But $K/F$ need not be Galois, because $K$ might not be a splitting field of a polynomial with coefficients in $F$.