Dimensions of Anisotropic Indefinite Quadratic Forms II To Andrei Suslin on the occasion of his 60th birthday

The u-invariant and the Hasse number ũ 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 ‘algebrogeometric’ characterization of fields with finite Hasse number. 2010 Mathematics Subject Classification: primary: 11E04; secondary: 11E10, 11E81, 14C25