Separable forms of $ {{\mathbb{G}}_m} $ -ACTIONS ON $ {{\mathbb{A}}^3} $

Abstract

Let k be a field of characteristic zero. We consider k-forms of \( {\mathbb G} \) m -actions on \( {\mathbb A} \) 3 and show that they are linearizable. In particular, \( {\mathbb G} \) m -actions on \( {\mathbb A} \) 3 are linearizable, and k-forms of \( {\mathbb A} \) 3 that admit an effective action of an infinite reductive group are trivial.