We present criteria for a pair of maps to constitute a quaternion-symbol equivalence (or a Hilbert-symbol equivalence if we deal with global function fields) expressed in terms of vanishing of the Clifford invariant. In principle, we prove that a local condition of a quaternion-symbol equivalence can be transcribed from the Brauer group to the Brauer-Wall group.

Abstract. Let F be a formally real field. Denote by G(F) and $G_t\left(F\right)$ the Grothen-dieck group of quadratic forms over F and its torsion subgroup, respectively. In this paper we study the structure of the factor group $G\left(F\right)/G_t\left(F\right)$. This reduced Grothendieck group is a free Abelian group. The main results of the paper describe some sets of generators for $G\left(F\right)/G_t\left(F\right)$, which in many cases allow us to find a basis for the group. Throughout the paper we use the language of the reduced theory of quadratic forms. In the final part...