Frobenius' Theorem on Division Algebras

TL;DR

This paper provides a simple, undergraduate-level proof of Frobenius' Theorem, which classifies all finite-dimensional real division algebras as only the reals, complexes, and quaternions.

Contribution

It offers a concise, accessible proof of Frobenius' Theorem using only standard undergraduate mathematics, simplifying understanding of the classification.

Findings

Real, complex, and quaternion algebras are the only finite-dimensional real division algebras.

The proof is accessible to undergraduates, avoiding advanced techniques.

The classification is complete and well-established in algebra.

Abstract

Frobenius' Theorem states that the only finite-dimensional real division algebras are the algebra of real numbers $\mathbb R$, the algebra of complex numbers $\mathbb C$, and the algebra of quaternions $\mathbb H$. We present a short proof which uses only standard undergraduate mathematics.