PlanetMath: composition algebras over $\mathbb{Q}$

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composition algebras over $\mathbb{Q}$

(Example)

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, $\CayDick{p}{\mathbb{Q}}$ for $p$ a prime number. This is sufficient to illustrate an infinite number of quadratic composition algebras. $\qedsymbol$

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

"composition algebras over $\mathbb{Q}$ " is owned by Algeboy.

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See Also: Hurwitz's theorem, Jacobson's theorem on composition algebras

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Cross-references: isometric, congruent, theory, complex number, rational numbers, quadratic forms, proofs, division algebras, number, infinite, sufficient, prime number, quadratic extension, composition algebras

This is version 3 of composition algebras over $\mathbb{Q}$ , born on 2007-06-23, modified 2007-08-15.
Object id is 9656, canonical name is CompositionAlgebrasOverMathbbQ.
Accessed 591 times total.

Classification:

AMS MSC:

17A75 (Nonassociative rings and algebras :: General nonassociative rings :: Composition algebras)

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