Universal quadratic form

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]

Examples
  • Over the real numbers, the form x2 in one variable is not universal, as it cannot represent negative numbers: the two-variable form x2 − y2 over R is universal.
  • Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form x2 + y2 + z2 + t2 − u2 over Z is universal.
  • Over a finite field, any non-singular quadratic form of dimension 2 or more is universal.[3]

Forms over the rational numbers

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]

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

References
  1. Lam (2005) p.10
  2. Rajwade (1993) p.146
  3. Lam (2005) p.36
  4. Serre (1973) p.43
  5. Serre (1973) p.37
  • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
  • Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
  • Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. Vol. 7. Springer-Verlag. ISBN 0-387-90040-3. Zbl 0256.12001.
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