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] 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.