# 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 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 $q\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(mod4)$. But if $p\equiv 3\phantom{\rule{veryverythickmathspace}{0ex}}(mod4)$ then this cannot be said.

Title | composition algebras over $\mathbb{Q}$ |
---|---|

Canonical name | CompositionAlgebrasOvermathbbQ |

Date of creation | 2013-03-22 17:18:29 |

Last modified on | 2013-03-22 17:18:29 |

Owner | Algeboy (12884) |

Last modified by | Algeboy (12884) |

Numerical id | 6 |

Author | Algeboy (12884) |

Entry type | Example |

Classification | msc 17A75 |

Related topic | HurwitzsTheorem |

Related topic | JacobsonsTheoremOnCompositionAlgebras |