The square class invariant (Q919379)

For a field F and a quadratic form Q over F, the square class invariant for Q is the function \(f_ Q: \dot F/\dot F^ 2\to {\mathbb{Z}}\) defined by \(f_ Q(a\dot F^ 2)=n\) if \(Q\cong n<a>+P\) and a is not represented by P. If F is non-formally real the level of F, s(F), is the smallest number of squares adding to -1. The following are proved: If F is non- pythagorean and s(F)\(\geq 4\) then \(f_ Q\) does not classify forms over F. F is a C-field if its anisotropic forms are uniquely determined by their value sets. If s(F)\(\leq 2\) and \(f_ Q\) classifies forms over F, then F is a C-field. If F is a C-field with \(s(F)=1\) or 2 and either u(F) or \(| \dot F/D(<1,1>)|\) is finite, then \(f_ Q\) classifies forms over F. If F is nonreal, there are more than two quaternion algebras over F, and \(u(F)=4\), then \(f_ Q\) and det do not classify forms over F.