Isotropy of quadratic forms and field invariants (Q2704199)

This is a very readable survey article about quadratic forms over fields. It is intended for non-specialists but also specialists will find interesting results or new proofs.NEWLINENEWLINENEWLINEThe first three sections contain basic material which mostly can be found in the books of Lam or Scharlau. In particular the \(u\)-invariant, Hasse number, level, Pythagoras number and length of a field \(F\) are introduced and studied. The main sections 4 and 5 outline the method of Merkurjev for the construction of fields with ``prescribed invariants'' e.g. \(u(F)=2n\) or \(p(F)=n\) for any \(n\in\mathbb{N}\). They are intimately related to the problem which quadratic forms \(\varphi\) become isotropic over the function field \(F(\psi)\) of another quadratic form \(\psi\). A beautiful theorem of the author (Theorem 4.4) states the following:NEWLINENEWLINENEWLINE``Let \(\varphi\) and \(\psi\) be anisotropic forms over \(F\) such that \(\dim\varphi\leq 2^n<\dim\psi\). Then \(\varphi\) stays anisotropic over \(F(\psi)\).''NEWLINENEWLINENEWLINEThe last section contains further examples and remarks, for instance the highly sophisticated construction of a field \(F\) with \(u(F)=9\) by Izhboldin. The bibliography has more than 80 items.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00036].