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Universal quadratic form

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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.

Contents

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.

    • 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). 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. One can conclude that all indefinite forms of dimension at least 4 over Q are universal.

    References

    Universal quadratic form Wikipedia
    (Text) CC BY-SA


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