Let L/K be a finite Galois extension of complete discrete valued fields of characteristic p. Assume that the induced residue field extension $k_L/k_K$ is separable. For an integer n ≥ 0, let $W_n{(}_L)$ denote the ring of Witt vectors of length n with coefficients in ${}_L$. We show that the proabelian group ${H^1(G,W_n{(}_L))}_{n\in \mathbb{N}}$ is zero. This is an equicharacteristic analogue of Hesselholt’s conjecture, which was proved before when the discrete valued fields are of mixed characteristic.

A Brückner-Vostokov formula for the Hilbert symbol of a formal group was established by Abrashkin under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ($\varphi ,\Gamma$)-modules and a cohomological interpretation of Abrashkin’s technique. To do this, we build ($\varphi ,\Gamma$)-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the...