Submission: 2009, Oct 26
Let F be a field of characteristic different from 2. The u-invariant and the Hasse number of a field F are classical and important field invariants pertaining to quadratic forms. These invariants measure the suprema of dimensions of anisotropic forms over F that satisfy certain additional properties. We prove new relations between these invariants and we give a new characterization of fields with finite Hasse number, the first one of its kind that uses intrinsic properties of quadratic forms and which, conjecturally, allows an "algebro-geometric" characterization of fields with finite Hasse number. We also construct various examples of fields with infinite Hasse number and prescribed finite values of u that satisfy additional properties pertaining to the space of orderings of the field.
2000 Mathematics Subject Classification: 11E10, 11E81, 14C25
Keywords and Phrases: quadratic form, Pfister form, Pfister neighbor, real field, ordering, strong approximation property, effective diagonalization, $u$-invariant, Hasse number, Pythagoras number, function field of a quadratic form, Rost correspondence, Rost projector
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