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 $x^2$ in one variable is not universal, as it cannot represent negative numbers: the two-variable form $x^2-y^2$ over $ℝ$ is universal.
• Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form $x^2+y^2+z^2+t^2-u^2$ over $\mathbb{Z}$ 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 $\mathbb{Q}$ if and only if it is universal over [[p-adic number|Template:Math]] for all primes $p$ (where we include $p=\mathrm{\infty }$, letting ${\mathbb{Q}}_{\mathrm{\infty }}$ denote $ℝ$). A form over $ℝ$ is universal if and only if it is not definite; a form over ${\mathbb{Q}}_p$ is universal if it has dimension at least 4. One can conclude that all indefinite forms of dimension at least 4 over $\mathbb{Q}$ are universal.

See also

• The 15 and 290 theorems give conditions for a quadratic form to represent all positive integers.

References

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