  Let $K$ be a complete discrete valued field of characteristic zero with
residue field $k_K$ of characteristic $p > 0$. Let $L/K$ be a finite Galois
extension with the Galois group $G$ and suppose that the induced extension of
residue fields $k_L/k_K$ is separable. In his paper, Hesselholt conjectured
that $H^1(G,W(\sO_L))$ is zero, where $\sO_L$ is the ring of integers of $L$
and $W(\sO_L)$ is the Witt ring of $\sO_L$ w.r.t. the prime $p$. He partially
proved this conjecture for a large class of extensions. In this paper, we prove
Hesselholt's conjecture for all Galois extensions.
