In mathematics, a universal quadratic form is a quadratic form over a ring that represents every element of the ring.Lam (2005) p.10 A non-singular form over a field which represents zero non-trivially is universal.Rajwade (1993) p.146
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.Lam (2005) p.36
Forms over the rational numbers The Hasse–Minkowski theorem implies that a form is universal over if and only if it is universal over for all primes (where we include , letting denote ).Serre (1973) p.43 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.Serre (1973) p.37 One can conclude that all indefinite forms of dimension at least 4 over are universal.
See also The 15 and 290 theorems give conditions for a quadratic form to represent all positive integers.
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