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.
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 , letting Q∞ denote R). 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. One can conclude that all indefinite forms of dimension at least 4 over Q are universal.