On, around, and beyond Frobenius' theorem on division algebras

Abstract

ABSTRACT Frobenius' Theorem states that, besides the fields of real and complex numbers, the algebra of quaternions is the only finite-dimensional real division algebra. We first give a short elementary proof of this theorem, then characterize finite-dimensional real algebras that contain either a copy of , a copy of , or a pair of anticommuting invertible elements through the dimensions of their (left) ideals, and finally consider the problem of lifting algebraic elements modulo ideals.