Dimensions of Anisotropic Indefinite Quadratic Forms Ii

Let F be a field of characteristic different from 2. The u-invariant and the Hasse number ũ 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.