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

The Hasse–Minkowski theorem implies that a form is universal over Q if and only if it is universal over Qp for all p. 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. One can conclude that all indefinite forms of dimension at least 4 over Q are universal.