Excellence of G_2 and F_4

TL;DR

This paper proves that algebraic groups of types G_2 and F_4 are excellent over any field with characteristic not 2 or 3, meaning their anisotropic kernels are defined over the base field for all extensions.

Contribution

It establishes the excellence of G_2 and F_4 groups over fields of characteristic not 2 or 3, extending understanding of their structural properties.

Findings

G_2 and F_4 groups are excellent over such fields

Anisotropic kernels are defined over the base field for all extensions

Results hold for fields with characteristic not 2 or 3

Abstract

A linear algebraic group G defined over a field k is said to be excellent if for every field extension L of k the anisotropic kernel of the group (G \otimes_k L) is defined over k. We prove that groups of type G_2 and F_4 are excellent over any field k of characteristic other than 2 and 3.