Similitudes over fields with I^4=0

TL;DR

This paper investigates the structure of similitudes over certain fields, extending previous results by relaxing conditions on the base field and Clifford invariants, with implications for quadratic forms and algebraic groups.

Contribution

It extends main results on similitudes and algebraic groups to broader classes of fields, relaxing conditions on the base field and Clifford invariants.

Findings

Characterization of R-equivalence classes over specific fields

Extension of previous results to new classes of fields

Insights into quadratic forms with nontrivial zeros

Abstract

This article studies the set of R-equivalence classes of the group of proper projective similitudes of an algebra with involution of the first kind. The main results concern base fields of characteristic different from 2 over which every 9-dimensional quadratic form has a nontrivial zero. This includes function fields of p-adic curves and extensions of transcendence degree 3 of C. Main results of [28] and [29] are extended by relaxing the condition on the base field as well as on the Clifford invariant for orthogonal involutions.