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

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

References

  1. Lam (2005) p.10
  2. Rajwade (1993) p.146
  3. Lam (2005) p.36
  4. 1 2 Serre (1973) p.43
  5. Serre (1973) p.37
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