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

• 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.

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 ). 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. 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.

References