An ultimate proof of Hoffmann–Totaro’s conjecture

Abstract

We prove the last open case of the conjecture on the possible values of the first isotropy index of an anisotropic quadratic form over a field. It was initially stated by Detlev Hoffmann for fields of characteristic $\ne 2$ and then extended to arbitrary characteristic by Burt Totaro. The initial statement was proven by the author in 2002. In characteristic $2$, the case of a totally singular quadratic form was done by Stephen Scully in 2015 and the nonsingular case by Eric Primozic in early 2019.