  Our investigation focuses on an additive analogue of the Bloch-Gabber-Kato
theorem which establishes a relation between the Milnor $K$-group of a field of
positive characteristic and a Galois cohomology group of the field. Extending
the Aritin-Schreier-Witt theory, we present an isomorphism from the Mackey
product associated with the Witt group and the multiplicative groups to a
Galois cohomology group. As a result, we give an expression for the torsion
subgroup of the Brauer group of a field, and more generally, the Kato homology
groups.
