An isomorphic classification of non-associative division octonion algebras

Question

A division octonion algebra over a field $F$ is a $8$-dimensional unital non-associative algebra $A$ over the field $F$, endowed with a quadratic form $N:A\times A\to F$ such that $N(xy)=N(x)N(y)$ and $N(x)\ne 0$ for every non-zero elements $x,y\in A$.

Wikipedia writes that two division octonion algebras $(A,N)$ and $(A',N')$ over a field $F$ are $F$-isomorphic if and only if their norms $N$, $N'$ are $F$-isomorphic. This reduces the problem of isomorphic classification of division octonion algebras to the problem of classification of their norms. Is there any additional information on such a classification?

Question. What is the number of non-isomorphic division octonion algebras over a given field? Can it be greater than one? Can it be infinite?

Because finite or algebraically closed fields admit no division octonion algebras, see the lecture notes of Konrad Voelkel.

Answer 1

Yes, the number of isomorphism classes can be greater than 1 and it can be infinite.

One good reference on octonions is the book [1] by Springer-Veldkamp. It gives the following example on page 22: If $K$ is a number field with $r$ real embeddings, then there are $2^r - 1$ isomorphism classes of octonion division algebras over $K$. This result goes back to Zorn [2].

To see that the number of octonion division algebras over a field can be infinite, here is another example. As Springer-Veldkamp explains on page 21, an octonion algebra can be specified by three nonzero elements $a, b, c$ in a field $F$ (which we assume for simplicity has characteristic different from 2). I’ll write $(a, b, c)$ for the corresponding octonion $F$-algebra. To align notation with other sources, my $a$ is what Springer-Veldkamp would call $-\alpha$, my $b$ is their $-\beta$, etc. With this convention, if $a = 1$, $b = 1$, or $c = 1$, then the octonion algebra $(a, b, c)$ is split but not necessarily conversely. In the language of Lam’s book on quadratic forms [3], the norm of the algebra $(a, b, c)$ is the Pfister form ${\ll} {-a}, -b, -c {\gg} = {\ll} \alpha, \beta, \gamma {\gg}.$

Write $F^{\times 2}$ for the group of squares in $F^\times$. In the following statement, $u$ and $v$ are indeterminates.

Proposition. For $K := F((u))((v))$:

1. If $x \in F^\times$ is not a square, then the octonion $K$-algebra $(u, v, x)$ is a division algebra.
2. If $1, x, y \in F^\times$ represent distinct classes in $F^\times/ F^{\times 2}$ then the octonion $K$-algebras $(u, v, x)$ and $(u, v, y)$ are division algebras that are not isomorphic to each other.

Proof. $K$ is a field complete with respect to a discrete valuation, the $v$-adic valuation, so we may apply the classic results in this setting, which can be found for example in section VI.1 of Lam’s book.

To prove claim 1, take the first residue form of the quadratic norm form ${\ll} {-u}, -v, -x {\gg}$ to find a Pfister quadratic form ${\ll} {-u}, -x {\gg}$ over $F((u))$. Repeating this procedure with the $u$-adic valuation in $F((u))$ we obtain a residue form ${\ll} -x {\gg}$ over $F$. Because $x$ is not a square in $F$, this form is anisotropic (Lam, Theorem I.3.2), and therefore the norm of the octonion algebra over $K$ is anisotropic (Lam, Proposition VI.1.9), so the octonion algebra is division (Springer-Veldkamp, Theorem 1.8.1).

As for claim 2: By claim 1 the two algebras are division. Repeating the argument in the previous paragraph, we find residue Pfister forms ${\ll} {-x} {\gg}$ and ${\ll} {-y} {\gg}$ over $F$ for each of the octonion norm forms. Because $xy$ is not a square, $-x$ and $-y$ represent distinct square classes. That is, the residue Pfister forms have distinct discriminants and therefore cannot be isomorphic. Therefore, the octonion norm forms are not isomorphic (because the first residue form is well defined), so the octonion algebras are not isomorphic (Springer-Veldkamp, Theorem 1.7.1).

Corollary. If $F^\times / F^{\times 2}$ is infinite, then $K$ as in the proposition has infinitely many octonion division algebras.

Proof. The group $F^\times / F^{\times 2}$ is a vector space over the field with two elements. Pick a basis of it, call the basis $X$ and note that it is infinite. For $x, y$ distinct elements of $X$, we may apply the proposition to see that the octonion $K$-algebras $(u, v, x)$ and $(u, v, y)$ are distinct and are division algebras. Therefore, the number of distinct octonion division algebras over $K$ is at least $|X|$.

Example. Take $F = \mathbb{Q}$. The vector space $\mathbb{Q}^\times / \mathbb{Q}^{\times 2}$ is infinite -- it has a basis consisting of the prime integers and $-1$. By the corollary, the field $K := F((u))((v))$ has infinitely many isomorphism classes of octonion division algebras.

References

[1]: Springer, Tonny A.; Veldkamp, Ferdinand D., Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics. Berlin: Springer (ISBN 3-540-66337-1/hbk). viii, 208 p. (2000). ZBL1087.17001.

[2]: Zorn, Max, Alternativkörper und quadratische Systeme, Abh. Math. Semin. Hamb. Univ. 9, 395-402 (1933). ZBL0007.05403.

[3]: Lam, T. Y., Introduction to quadratic forms over fields, Graduate Studies in Mathematics 67. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-1095-2/hbk). xxi, 550 p. (2005). ZBL1068.11023.