In the case of a totally ramified extension of a complete field, then we know that there exist a maximum tamely ramified subextension, which is of degree coprime to $p$, and from that onward there are only wild ramification. Hence, if the extension is Galois, the inertia group (which is the whole Galois group), contains a normal $p$-Sylow subgroup.
I am not sure why every book I read on this only deal with local field (ie. complete valuation field). Is it simply not true for global field? I thought the same might be true for global field by some application of Krasner's lemma. But then again, none of the books mentioned that, so I am suspicious.