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arXiv · 1912.07846

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

Matej Brešar·Victor S. Shulman
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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.

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Matej Brešar, Victor S. Shulman. 2019-12-17. On, Around, and Beyond Frobenius' Theorem on Division Algebras. https://arxiv.org/abs/1912.07846

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