ABSTRACT

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

中文翻译：

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关于除法代数的弗罗贝尼乌斯定理，围绕和超越

摘要

Frobenius 定理指出，除了实数和复数的域外，四元数的代数$\mathbb{H}$是唯一的有限维实除代数。我们首先给出这个定理的一个简短的基本证明，然后刻画包含$\mathbb{C}$, 一份$\mathbb{H}$, 或一对反对易可逆元通过其（左）理想的维数，最后考虑提升代数元素模理想的问题。