# 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]}

## Examples

- Over the real numbers, the form
*x*^{2}in one variable is not universal, as it cannot represent negative numbers: the two-variable form*x*^{2}−*y*^{2}over**R**is universal. - Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form
*x*^{2}+*y*^{2}+*z*^{2}+*t*^{2}−*u*^{2}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 **Q**_{p} 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 **Q**_{p} 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

- 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.

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