Quaternary quadratic form

A quadratic form in four variables. A quaternary quadratic form over a field $ F $ is related to the algebra of quaternions (cf. Quaternion) over the same field. Namely, corresponding to the algebra with basis $ [ 1 , i _ {1} , i _ {2} , i _ {3} ] $, $ i _ {1} ^ {2} = - a _ {1} \in F $, $ i _ {2} ^ {2} = - a _ {2} \in F $, and $ i _ {1} i _ {2} = - i _ {2} i _ {1} = i _ {3} $, is the quaternary quadratic form which is the norm of the quaternion,

$$ q ( x _ {0} , x _ {1} , x _ {2} , x _ {3} ) = \ N ( x _ {0} + x _ {1} i _ {1} + x _ {2} i _ {2} + x _ {3} i _ {3} ) = $$

$$ = \ x _ {0} ^ {2} + a _ {1} x _ {1} ^ {2} + a _ {2} x _ {2} ^ {2} + a _ {1} a _ {2} x _ {3} ^ {2} . $$

For quaternary quadratic forms corresponding to quaternion algebras, and only for these, composition of quaternary quadratic forms is defined:

$$ q ( x) q ( y) = q ( z) , $$

where the coordinates of the vector $ z $ are bilinear forms in $ x $ and $ y $. Composition of this kind is possible only for quadratic forms in two, four and eight variables.

Comments

The last-mentioned result is known as Hurwitz's theorem; see Quadratic form.