Galois group of a non-separable polynomial

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