composition algebras over

Theorem 1.

There are infinitely many composition algebrasMathworldPlanetmath over Q.


Every quadratic extension of is a distinct composition algebra. For example, (p) for p a prime numberMathworldPlanetmath. 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 formsMathworldPlanetmath of the form:


for rational numbers p and q. Such questions can involve complex numberMathworldPlanetmathPlanetmath theory as for instance, if p is a prime congruentMathworldPlanetmath 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 q1(mod4). But if p3(mod4) then this cannot be said.

Title composition algebras over
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