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We prove a local analogue of a theorem of J. Martinet about the absolute norm of the relative discriminant ideal of an extension of number fields. The result can be seen as a statement about $2$-primary units. We also prove a similar statement about the absolute norms of $p$-primary units, for all primes $p$.

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.