Inversion invariant additive subgroups of division rings

Abstract

We characterize the inversion invariant additive subgroups of any field, and, more generally, those of a division ring (apart from division rings of characteristic 2).We also show how a classical identity of Hua provides a bridge between this problem and Jordan algebras.Answering a question of Dan Mauldin, we characterize in this note the inversion invariant additive subgroups of a field.We show in Section 2 that aside from the case of imperfect fields of characteristic 2, a nonzero subgroup that is inversion invariant is either a subfield or the set of trace-zero elements in a subfield with respect to an automorphism of order 2. A key ingredient in the proof is a simple, classical identity of Hua, through which we can bring to bear known results about Jordan algebras and Jordan triple systems.We also solve in Section 1 the same problem for division rings of characteristic not 2.