In my notes, I am given the definition of the Galois group of a polynomial only in the case when the polynomial is separable (if $f$ is a separable polynomial over $K$ with splitting field $L$, then $\mathrm{Gal}(f) = \mathrm{Gal}(L/K)$). This makes sense, since $L$ is a splitting field implies normality, and an extension generated by separable elements is separable, so $L$ is necessarily a Galois extension of $K$.
I should probably tell you that my definition of separable is the one which involves the irreducible factors of the polynomial having distinct roots in a splitting field (I'm aware there's another definition, and that they coincide when defining a separable extension).
When I look around online, I see that the Galois group is defined for any polynomial as the Galois group of its splitting field. Why is the splitting field necessarily Galois?
Thanks
Best Answer
Field extensions in characteristic 0 are always separable and finite extensions of finite fields are always separable as well.
Only when infinite fields of characteristic $p>0$ are involved you can encounter inseparable extensions.
Maybe the online sources you mention only look at one of the settings where separability is always true. It is also possible that the authors use the term "Galois group" even in non-Galois situations where others only use the term "automorphism group".