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Is there a classification of the $p$-adic normed division algebras?

A normed division algebra over $\mathbb{R}$ is a pair $(A,\lVert{-}\rVert)$ with $A$ an $\mathbb{R}$-algebra with a unit $1_A$; $\lVert{-}\rVert\colon A\to\mathbb{R}_{\geq0}$ a norm on $A$; such ...
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