The present article comprises the lecture notes of a mini-course given by the author at the Université d’Artois, Lens, June 26–28, 2000. In it, the author describes some of the work which O. Izhboldin completed shortly before his tragic and untimely death. The first part deals with results on virtual Pfister neighbors and the first Witt index, and it is based more or less on Izhboldin’s paper in the same volume [Lect. Notes Math. 1835, 131–142 (2004; Zbl 1053.11033)].
The second part of the article explains the construction of fields with \(u\)-invariant equal to \(9\) due to O. Izhboldin [Ann. Math. (2) 154, No. 3, 529–587 (2001; Zbl 0998.11015)].
The article gives a very readable overview of the results in these two papers. With most of the proofs worked out in detail or at least thoroughly sketched, sometimes supplemented by various alternative arguments, the structure of the proofs and the ideas behind them become quite clear. This article is therefore quite suitable as an introduction for those readers who want to learn more about these beautiful and important results from the algebraic theory of quadratic forms and how powerful methods drawn from the theories of Chow groups and unramified cohomology are put to use to prove them.
For the entire collection see [Zbl 1034.14001].