Universal quadratic form

In mathematics, a universal quadratic form is a quadratic form over a ring that represents every element of the ring.[1] A non-singular form over a field which represents zero non-trivially is universal.[2]

## Contents

* 1 Examples
* 2 Forms over the rational numbers
* 3 See also
* 4 References

## Examples

* Over the real numbers, the form x2 in one variable is not universal, as it cannot represent negative numbers: the two-variable form x2 − y2 over R is universal.
* Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form x2 \+ y2 \+ z2 \+ t2 − u2 over Z is universal.
* Over a finite field, any non-singular quadratic form of dimension 2 or more is universal.[3]

## Forms over the rational numbers

The Hasse–Minkowski theorem implies that a form is universal over Q if and only if it is universal over Qp for all p (where we include p = ∞, letting Q∞ denote R).[4] A form over R is universal if and only if it is not definite; a form over Qp is universal if it has dimension at least 4.[5] One can conclude that all indefinite forms of dimension at least 4 over Q are universal.[4]

## See also

* The 15 and 290 theorems give conditions for a quadratic form to represent all positive integers.

## References

1. ↑ Lam (2005) p.10
2. ↑ Rajwade (1993) p.146
3. ↑ Lam (2005) p.36
4. ↑ 4.0 4.1 Serre (1973) p.43
5. ↑ Serre (1973) p.37

* Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2.
* Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. 171. Cambridge University Press. ISBN 0-521-42668-5.
* Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. 7. Springer-Verlag. ISBN 0-387-90040-3. https://archive.org/details/courseinarithmet00serr.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Universal quadratic form. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Universal_quadratic_form&oldid=199583"