Ramified primes, as defined by Neukirch

I took a look at the corresponding section in the book. Although the end goal is of course to apply all this to number fields, this is actually not the assumption in that section. Neukirch works at the level of generality of Dedekind domain, not rings of integers of number fields. And in that more general case, the residue fields don't have to be finite, and the inertia extension does not have to be separable.