On, Around, and Beyond Frobenius' Theorem on Division Algebras

TL;DR

This paper provides a concise proof of Frobenius' Theorem, characterizes certain finite-dimensional real algebras containing complex or quaternionic structures, and explores lifting algebraic elements modulo ideals.

Contribution

It offers an elementary proof of Frobenius' Theorem and characterizes algebras with complex or quaternionic substructures based on ideal dimensions.

Findings

Elementary proof of Frobenius' Theorem

Characterization of algebras with complex or quaternionic subalgebras

Analysis of lifting algebraic elements modulo ideals

Abstract

Frobenius' Theorem states that the algebra of quaternions $\mathbb H$ is, besides the fields of real and complex numbers, 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 $\mathbb C$, a copy of $\mathbb H$, 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.