composition algebras over
Theorem 1.
There are infinitely many composition algebras over .
Proof.
Every quadratic extension of is a distinct composition algebra. For example,
for 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:
for rational numbers and . Such questions can involve complex number theory as
for instance, if is a prime congruent
to modulo then
is isometric to and thus is isometric to for
any other prime . But if 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 |