Galois cohomology in degree 3 of function fields of curves over number fields

Abstract

Let k be a field of characteristic not equal to 2. For n≥1, let Hⁿ(k, Z/Z) denote the nth Galois Cohomology group. The classical Tate's lemma asserts that if k is a number field then given finitely many elements , α₁, ..., α_n(k, Z/2), there exist a, b₁..., b_n ε such that a_i= (a)∪(b_i), where for any λε k*, (λ) denotes the image of k* in H¹(k, Z/2). In this paper we prove a higher dimensional analogue of the Tate's lemma.