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 H n(k, Z /2) denote the nth Galois Cohomology group. The classical Tate's lemma asserts that if k is a number field then given finitely many elements α 1,⋯,α n∈H 2(k, Z /2) , there exist a,b 1,⋯,b n∈k ∗ such that α i =( a)∪( b i ), where for any λ∈k ∗ , ( λ) denotes the image of k ∗ in H 1(k, Z /2) . In this paper we prove a higher dimensional analogue of the Tate's lemma.