I'm trying to understand Theorem I.9.4 from Neukirch's Algebraic Number Theory (page 56).
First he proves that $\kappa(\mathfrak{P})\mid \kappa(\mathfrak{p})$ is a normal extension, which is fine with me.
But I'm confused with the proof that $G_{\mathfrak{P}}\to G(\kappa(\mathfrak{P})\mid \kappa(\mathfrak{p}))$ is surjective.
He talks about a "maximal separable subextension" of $\kappa(\mathfrak{P})\mid \kappa(\mathfrak{p})$. But isn't the extension itself separable? I mean, $\kappa(\mathfrak{p}):=\mathfrak{o}/\mathfrak{p}$ is a finite field (because $\mathfrak{o}$ is Dedekind), therefore a perfect field, so any finite extension of it should be separable, right?
Besides, aren't we assuming $\kappa(\mathfrak{P})\mid \kappa(\mathfrak{p})$ is Galois when we write "$G(\kappa(\mathfrak{P})\mid \kappa(\mathfrak{p}))$"?
One more thing: he writes $G(\kappa(\mathfrak{P})\mid \kappa(\mathfrak{p}))=G(\kappa(\mathfrak{p})(\overline{\theta})\mid \kappa(\mathfrak{p}))$, where $\overline{\theta}$ is a primitive element for the "maximal separable subextension". Why is that true?