In their book Octonions, Jordan Algebras and Exceptional groups Springer and Veldkamp have a subsection called 'Classification over special fields' in which they describe the number of division and split quaternion and octonion algebras (and how to find them) over the following fields (as quoted from the book):

Algebraically closed fields, the reals, finite fields, complete, discretely valuated fields with finite residue fields and for algebraic number fields.

For finite fields, all quaternion and octonion algebras are split (by Wedderburn's Little Theorem). For complete, discretely valuated fields with finite residue fields, there is exactly one isomorphism class of quaternion division algebra (in addition to the split quaternion algebra that always exists) but all octonion algebras are split again.

But there is a class of fields `in between' these two classes that is not on the list: NOT-complete, discretely valuated fields with finite residue fields, or more concretely the fields $\mathbb{F}_p(t)$ for $p > 2$.

Does anyone know if division octonion algebras over these fields exist?

  • 1
    $\begingroup$ An octonion algebra $A$ over an arbitrary field (allowing characteristic 2) is split if and only if its norm form $N:A \to k$ defined by $a \mapsto a a^*$, a non-degenerate quadratic space of dimension 8, is isotropic (for a proof, see 1.8.1 in the book Octonions, Jordan algebras, and exceptional groups by Springer and Veldkamp). But over any global function field $k$ a non-degenerate quadratic space of dimension $\ge 5$ is isotropic due to the Hasse-Minkowski theorem over $k$ since the same holds over any non-archimedean local field. Thus, over such $k$ any octonion algebra is split. $\endgroup$ – nfdc23 Oct 11 '17 at 15:25

The complete classification for fields of characteristic unequal to 2 is given in Section 8 of Serre's paper

  • J.-P. Serre. Cohomologie galoisienne : progrès et problèmes. Séminaire Bourbaki, Volume 36 (1993-1994) , Talk no. 783 , p. 229-257.

The main result is that octonion algebras over a field $F$ are classified by decomposable classes in $H^3(F,\mathbb{Z}/2\mathbb{Z})$, or alternatively by Pfister 3-forms over the field $F$. Note also that an octonion algebra is either split or division.

In particular, if the 2-cohomological dimension of the field is at most 2, then there are no nonsplit octonion algebras. Function fields in one variable over finite fields have cohomological dimension $2$ (p. 85 of Serre's Galois cohomology book), so there are no nonsplit octonion algebras in that case.

For a cohomological dimension 3 case: there is an explicit classification of octonion algebras over $K(T)$ where $K$ is a $p$-adic field in Serre's article, section 8.3.


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