#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Andrei A. Suslin's Sixtieth Birthday (2010), 251-265

** Detlev W. Hoffmann **
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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