## Dimensions of anisotropic indefinite quadratic forms. II.

### Summary

The $u$-invariant and the Hasse number $\tilde u$ 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.

### Mathematics Subject Classification

11E04, 11E10, 11E81, 12D15, 14C25

### Keywords/Phrases

Pfister neighbor, formally real field, $u$-invariant, Hasse number, Pythagoras number, Rost projector