Does all splitting field have characteristic 0?
This may be a bad question, but I am wondering because the author of a book I am reading summarises a lot of properties when he is about to start with Galois theory, one of the properties is:
If E is a finite extension of F and is a separable splitting field over F, then $|G(E/F)|=\{E:F\}=[E:F]$.
However in the relevant chapter I can not see this proved, the closest thing I find is:
We have completed our aim, which was to show that field of characteristic 0 and finite fields have only separable finite extensions, that is, these fields are perfect. For finite extensions E of such fields F, we then have $[E:F]=\{E:F\}$.
Does the first sentence in some way follow from the second? It would if I could prove that all splitting fields are are either finite or have characteristic 0?