composition algebras over $\mathbb{Q}$

Theorem 1.

There are infinitely many composition algebras over $\mathrm{Q}$.

Proof.

Every quadratic extension of $\mathbb{Q}$ is a distinct composition algebra. For example,$\left(\frac{p}{\mathbb{Q}}\right)$ for $p$ a prime number. This is sufficient to illustrate an infinitenumber of quadratic composition algebras.∎

The other families of composition algebras also have an infinite number of non-isomorphicdivision algebras though the proofs are more involved. It suffices to show providean infinite family of non-isometric quadratic forms of the form:

$$N_{p,q}(a,b,c,d)=a^2-b^{2}p-c^{2}q+d^{2}pq$$

for rational numbers $p$ and $q$. Such questions can involve complex number theory asfor instance, if $p$ is a prime congruent to $1$ modulo $4$ then$N_{-1,-p}$ is isometric to $N_{-1,-1}$ and thus $N_{-1,-p}$ is isometric to $N_{-1,-q}$ forany other prime $q\equiv 1(mod4)$. But if $p\equiv 3(mod4)$ then this cannot be said.