Ramification Theory for Local Field with Imperfect Residue Field

Definition 1.3. The most natural way to define higher ramification subgroups of the Galois group G is due by Hilbert: g ∈ Ga if and only if vl(gx− x) ≥ a + 1, ∀x ∈ Ol. Indeed, G−1 = G, G0 = I is the inertia subgroup, and G1 = W is the wild inertia subgroup. However, there is a disadvantage of this. Namely, it does not respect quotient and hence it does not give a filtration on the absolute Galois group Gk. Herbrand defined an ad hoc looking function φ and gave G an upper numbering filtration, which does extend to Gk. (We will give a working definition later.) Definition 1.4. We set Gu = Gdue for u ∈ [−1,∞). Define