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I have read that the only finite-dimesional division algebras over the real numbers have dimensions 1, 2, 4, and 8. Are there any infinite-dimensional division algebras over the real numbers?

mathlander
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    Yes, take the field of rational functions. This works over any field. – Qiaochu Yuan Oct 26 '22 at 22:46
  • How exactly do rational functions form an algebra over the real numbers? – mathlander Oct 26 '22 at 22:55
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    To be clear, I mean the field $\mathbb{R}(t)$, the fraction field of the ring $\mathbb{R}[t]$ of polynomials in one variable over $\mathbb{R}$. The $\mathbb{R}$-algebra structure comes from the inclusion of constant functions. Again, this works with $\mathbb{R}$ replaced by any field. – Qiaochu Yuan Oct 26 '22 at 23:55

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