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. [4]
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. [4]