Meyer's theorem

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In mathematics, Meyer's theorem on quadratic forms states that a quadratic form Q over the rational numbers in at least five variables represents zero if and only if it is an indefinite form.^[1]

That is,

Q(v) = 0

for some non-zero rational vector v if and only if it does for some non-zero real vector.

By applying the Hasse-Minkowski theorem, one may find that the statement is equivalent to the assertion that such a form always represents zero for each p-adic field.

Meyer's theorem is best possible with respect to the number of variables since there are indefinite quadratic forms Q over the rational numbers in four variables which do not represent zero. No perfect square is congruent to 3 modulo 4, which shows that

Q(x₁,x₂,x₃,x₄) = x₁² + x₂² - p(x₃² + x₄²),

provides examples of such forms when p is a prime number congruent to 3 modulo 4.

[edit] References

1. ^ A. Meyer, Mathematische Mittheilungen, Vierteljahrschrift der Naturforschenden Gesellschaft in Zürich, 29, 209-222, (1884)

• Husemoller, D. and J. Milnor. "Symmetric Bilinear Forms." Ergebnisse der Mathematik und ihrer Grenzgebiete Band 73 (1973).