Difference between revisions of "Quaternary quadratic form"
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| + | $#C+1 = 11 : ~/encyclopedia/old_files/data/Q076/Q.0706760 Quaternary quadratic form | ||
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| − | + | A [[Quadratic form|quadratic form]] in four variables. A quaternary quadratic form over a field $ F $ | |
| + | is related to the algebra of quaternions (cf. [[Quaternion|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==== | ====Comments==== | ||
The last-mentioned result is known as Hurwitz's theorem; see [[Quadratic form|Quadratic form]]. | The last-mentioned result is known as Hurwitz's theorem; see [[Quadratic form|Quadratic form]]. | ||
Latest revision as of 08:09, 6 June 2020
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.
Quaternary quadratic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quaternary_quadratic_form&oldid=18432