Universal quadratic form

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

Examples

Over the real numbers, the form in one variable is not universal, as it cannot represent negative numbers: the two-variable form over is universal.
Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form over is universal.
Over a finite field, any non-singular quadratic form of dimension 2 or more is universal.

Forms over the rational numbers

The Hasse–Minkowski theorem implies that a form is universal over if and only if it is universal over [p-adic number|] for all primes . A form over is universal if and only if it is not definite; a form over is universal if it has dimension at least 4. One can conclude that all indefinite forms of dimension at least 4 over are universal.