Grothendieck and Witt groups in the reduced theory of quadratic forms

Abstract. Let F be a formally real field. Denote by G(F) and ${\displaystyle G_t\left(F\right)}$ 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 ${\displaystyle G\frac{F}{G_t}\left(F\right)}$. This reduced Grothendieck group is a free Abelian group. The main results of the paper describe some sets of generators for ${\displaystyle G\frac{F}{G_t}\left(F\right)}$, 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 of the paper wo apply the results to determine completely the structure of the reduced Grothendieck group ${\displaystyle G\frac{F}{G_t}\left(F\right)}$ for all fields with |g(F)| ≤ 16, where g(F) is the factor group F*lT(F), T (F) being the subgroup of all totally positive elements of F. All the results concerning Grothendieck groups have their counter-parts for Witt groups and we also state and prove the results iu that case.