# composition algebras over $\mathbb{Q}$

###### Theorem 1.

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

###### 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 infinite number of quadratic composition algebras. ∎

The other families of composition algebras also have an infinite number of non-isomorphic division algebras though the proofs are more involved. It suffices to show provide an 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 as for 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 $N_{-1,-q}$ for any other prime $q\equiv 1\pmod{4}$. But if $p\equiv 3\pmod{4}$ then this cannot be said.

Title composition algebras over $\mathbb{Q}$ CompositionAlgebrasOvermathbbQ 2013-03-22 17:18:29 2013-03-22 17:18:29 Algeboy (12884) Algeboy (12884) 6 Algeboy (12884) Example msc 17A75 HurwitzsTheorem JacobsonsTheoremOnCompositionAlgebras