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Let $k$ be a non-archimedean local field, for example, a $p$-adic field (a finite extension of the filed ${\Bbb Q}_p$ of $p$-adic numbers). It is well known that there is a canonical isomorphism $${\rm inv}\colon {\rm Br}\,k\overset\sim\longrightarrow{\Bbb Q}/{\Bbb Z}.$$ Let $D$ denote the division algebra of dimension 9 over $k$ with invariant $\frac13$.

Question. How can one explicitly describe the multiplication law in $D$ ?

Motivation. From the multiplication law in $D$, I can obtain the commutation law in the 8-dimensional Lie algebra ${\frak g}={\frak sl}(1,D)$. From $\frak g$, I can obtain an explicit trilinear alternating form on the 8-dimensional space $\frak g$: $$(x,y,z)\mapsto ([x,y],z)\quad\text{for }\,x,y,z\in{\frak g},$$ where $(\,,\,)$ denotes the Killing form. This is a $k$-form of a generic alternating trilinear form on $k^8$.

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    $\begingroup$ If $K/k$ is the unramified extension of degree 3, and $\pi$ a uniformiser, then $D$ is isomorphic to the cyclic algebra $(K/k,\pi)$. Is that what you're looking for? $\endgroup$
    – Martin Bright
    Commented Mar 16, 2021 at 9:21
  • $\begingroup$ @MartinBright: Yes! Many thanks! Could you please give a reference? $\endgroup$
    – Mikhail Borovoi
    Commented Mar 16, 2021 at 9:31
  • $\begingroup$ @MartinBright: What about the multiplication law in a division algebra of dimension 9 with given local invariants over a global field $k$ ? $\endgroup$
    – Mikhail Borovoi
    Commented Mar 16, 2021 at 9:35
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    $\begingroup$ Over a global field any division algebra is cyclic; this is one of the main theorems of class field theory (See Theorem 1.5.36 of Poonen - Rational Points over Varieties). The proof looks constructive so this may help. $\endgroup$
    – Daniel Loughran
    Commented Mar 16, 2021 at 10:40
  • $\begingroup$ @DanielLoughran: Thank you! I will try to ask a new question about division algebras over global fields. $\endgroup$
    – Mikhail Borovoi
    Commented Mar 16, 2021 at 11:01

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