There are the complex p-adic numbers. But what is the p-adic analogue of the Cayley–Dickson construction? Or more important: What is the p-adic analogue of the octonions? It would be nice if the (unit)-multiplication table of such a p-adic analogue corresponds to the projective plane over the finite field with p elements. Is there any interesting mathematical structure with such a multiplication table for a general p? Is there any research work about this?
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2$\begingroup$ The Cayley–Dickson construction makes sense starting from any (possibly non-associative) $*$-algebra over a ring: en.wikipedia.org/wiki/… so presumably one would need to take the $\mathbb{C}_p$ and just run the machine. I'm not sure what you mean by the next bit. $\endgroup$– David Roberts ♦Nov 17, 2021 at 0:59
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$\begingroup$ @DavidRoberts Surely this is a reference to the relation between the multiplication table of the standard basis of the octonions and the geometry of the projective plane over $\mathbb F_2$ (there is one non-$1$ basis vector for each point, and $e_i \cdot e_j = \pm e_k$ where $i,j,k$ are colinear.) I would guess division algebras with a similar relationship to projective planes over larger finite fields are known not to exist, but this isn't based on any hard evidence. $\endgroup$– Will SawinNov 19, 2021 at 3:35
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$\begingroup$ @WillSawin aha, that's what "(unit)-multiplication table" means! $\endgroup$– David Roberts ♦Nov 19, 2021 at 7:07
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3$\begingroup$ Already the issue of quaternion algebras extending the $p$-adics is subtle, see here: en.wikipedia.org/wiki/Quaternion_algebra#Classification. The reference here mathoverflow.net/a/283222/4177 might be of some use $\endgroup$– David Roberts ♦Nov 19, 2021 at 7:12
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2$\begingroup$ I never heard "complex $p$-adic numbers" for $\mathbf{C}_p$ although I now can see Planetmath advertises this definition. This is only partially an analogy. For $p$ such that $-1$ is not a square, one could also define "complex $p$-adic numbers" as $\mathbf{Q}_p[\sqrt{-1}]$. Or even any quadratic extension of $\mathbf{Q}_p$. This is closer to question of asking about a 8-dimensional non-associative algebra with given properties. $\endgroup$– YCorNov 20, 2021 at 11:21
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Defining and classifying the octonion algebras (composition algebras of dimension $8$) over fields $k$, or, in more sophisticated terms, computing the Galois cohomology set $H^1(k, G_2)$, is the topic of the book Octonions, Jordan Algebras and Exceptional Groups by T. A. Springer & F. D. Veldkamp (2000, appropriately published by Springer). Among other things, after properly defining what this means, they explain (§1.10(vi), page 22) that over the $p$-adic fields, just like over the finite fields or totally imaginary number fields, every octonion algebra is split, meaning it is isomorphic to the obvious one.
These notes by P. Gille discussing the more general problem of the classification of octonion algebras over rings, might also be worth looking at.
