Let K be a complete discrete valuation field of mixed characteristic and k be its residue field of prime characteristic p > 0. We assume that [k: kp] = ph < ∞. Let GK be the absolute Galois group of K and R be a Banach algebra over Cp:=K̄̂ with a continuous action of GK. When k is perfect (i.e. h = 0), Sen studied the Galois cohomology H1(GK, R*) and Sen's operator associated to each class (Sen Ann Math 127:647-661, 1988). In this paper we generalize Sen's theory to the case h ≥ 0 by using Brinon's theory (Brinon Math Ann 327:793-813, 2003). We also give another formulation of Brinon's theorem (à la Colmez).
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