Pawel Gladki - Institute of Mathematics, Faculty of Mathematics, Physics and Chemistry, University of Silesia
Krzysztof Worytkiewicz - Laboratorire de Math&#۰۳۹;{e}matiques, Universit&#۰۳۹;{e} Savoie Mont Blanc, B^{a}timent Le Chablais, Campus Scientifique, ۷۳۳۷۶ Le Bourget du Lac, France.

This paper introduces an approach to the axiomatic theory of quadratic forms based on presentable partially ordered sets, that is partially ordered sets subject to additional conditions which amount to a strong form of local presentability. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of quadratically\  presentable\  fields, that is, fields equipped with a presentable partial order inequationaly compatible with the algebraic operations. In particular, Witt rings of symmetric bilinear forms over fields of arbitrary characteristics are isomorphic to Witt rings of suitably built quadratically presentable fields.

Quadratically presentable fields, Witt rings, hyperfields, quadratic forms