This question is in reference to an answer I received from a previous question.
The answer talks about a cubic polynomial $p(x)$ over a (possibly non-perfect) field $F$ which factors over a splitting field as $p(x) = (x-\alpha)^2(x-\beta)$, where $\alpha$ and $\beta$ are not neccesarily distinct, and we are trying to determine if $\alpha$ and $\beta$ have to be in $F$.
My main hangup is the answer talks about the Galois group of $p(x)$, but Dummit only defines the Galois group of a separable polynomial, and only makes an extension of the definition to inseparable polynomials in the case of polynomials over perfect fields. So what is meant by the Galois group of an inseparable polynomial over a non-perfect field? Is it just the automorphism group of the splitting field $K$ of $p(x)$ over $F$, even if $K/F$ is not Galois?