Dimensions of Anisotropic Indefinite Quadratic Forms II
The $u$-invariant and the Hasse number $\hn$ of a field $F$ of characteristic not $2$ are classical and important field invariants pertaining to quadratic forms. They measure the suprema of dimensions of anisotropic forms over $F$ that satisfy certain additional properties. We prove new relations between these invariants and a new characterization of fields with finite Hasse number (resp. finite $u$-invariant for nonreal fields), 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.
2010 Mathematics Subject Classification: primary: 11E04; secondary: 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, Rost correspondence, Rost projector
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