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Quadratic form schemes determined by Hermitian forms. (English) Zbl 0621.10016

A quadratic form scheme is a triple \(S=(g,-1,d)\) where g is an elementary 2-group, -1 is in g, \(d: g\to \{subgroups\) of \(g\}\) and certain axioms hold. These axioms imitate the behavior of quadratic forms over a field F of characteristic \(\neq 2\), using \(g=F^{\times}/F^{\times 2}\), \(-1=(- 1)F^{\times 2}\), and \(d(a)=D_ F(<1,a>)/F^{\times 2}\) (the set of square classes represented by the binary quadratic form \(<1,a>)\). It is unknown whether every scheme S does arise this way from a field. The author shows that Hermitian forms over a division ring with involution also provide examples of a quadratic form scheme.
Reviewer: D.B.Shapiro

MSC:

11E16 General binary quadratic forms
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