Number Theory – p-Adic Analogue of Octonions

Question

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?

Answer 1

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.