ABSTRACT
Let K be a field of characteristic = 2, and let GK be Gal(Ks/K), where Ks is a separable closure of K. We will be interested in the Galois cohomology of K modulo 2; for brevity, let us denote the cohomology groups Hi(GK ,Z/2Z) by Hi(GK). In the case where i = 1, 2, Kummer theory provides the following interpretation of Hi(GK):
H1(GK) = K∗/K∗2,
H2(GK) = Br2(K),
where Br2(K) is the 2-torsion in the Brauer group of K. If a ∈ K∗, we denote by (a) the corresponding element of H1(GK). The cup-product
(a)(b) ∈ H2(GK)
corresponds to the class of the quaternion algebra (a, b) in Br2(K) defined by
i2 = a, j2 = b, ij = −ji.
