  Let F be a field of characteristic different from 2. The u-invariant and the
Hasse number 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 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. We also construct to each positive
integer n a real field F such such that the Hasse number is 2^{n+1} and such
that each quadratic form over F of dimension 1+2^n is a Pfister neighbor. These
results were announced (without proof) in the article "Dimensions of
anisotropic indefinite quadratic forms II" by the present author.
