Abstract
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.
Highlights
In this work we approach the axiomatic theory of quadratic forms by generalising the underlying principles of hyperrings [16] to certain partial orders we call presentable
Presentable posets generalise the behaviour of pierced powersets, that is powersets excluding the empty set with order given by inclusion
The presentable field induced by this hyperfield is quadratically presentable, and its Witt ring in our sense is isomorphic to its standard Witt ring
Summary
In this work we approach the axiomatic theory of quadratic forms by generalising the underlying principles of hyperrings [16] to certain partial orders we call presentable. We exhibit presentable groups, rings and fields arising in a natural way from hypergroups, hyperrings and hyperfields, respectively. This provides the main link between our theory and already existing axiomatic theories of quadratic forms. The presentable field induced by this hyperfield is quadratically presentable, and its Witt ring in our sense is isomorphic to its standard Witt ring What makes this construction of interest is the fact that it uniformely works for fields of both characteristic 2 and = 2. The techniques here heavily rely on the connection between presentable algebras and hyperalgebras
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