# Meyer's theorem

In number theory, **Meyer's theorem** on quadratic forms states that an indefinite quadratic form *Q* in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation

*Q*(*x*) = 0

has a non-zero real solution, then it has a non-zero rational solution (the converse is obvious). By clearing the denominators, an integral solution *x* may also be found.

Meyer's theorem is usually deduced from the Hasse–Minkowski theorem (which was proved later) and the following statement:

- A rational quadratic form in five or more variables represents zero over the field
**Q**_{p}of the p-adic numbers for all*p*.

Meyer's theorem is best possible with respect to the number of variables: there are indefinite rational quadratic forms *Q* in four variables which do not represent zero. One family of examples is given by

*Q*(*x*_{1},*x*_{2},*x*_{3},*x*_{4}) =*x*_{1}^{2}+*x*_{2}^{2}−*p*(*x*_{3}^{2}+*x*_{4}^{2}),

where *p* is a prime number that is congruent to 3 modulo 4. This can be proved by the method of infinite descent using the fact that if the sum of two perfect squares is divisible by such a *p* then each summand is divisible by *p*.

## See also

## References

- Meyer, A. (1884). "Mathematische Mittheilungen".
*Vierteljahrschrift der Naturforschenden Gesellschaft in Zürich***29**: 209–222. - Milnor, J.; Husemoller, D. (1973).
*Symmetric Bilinear Forms*. Ergebnisse der Mathematik und ihrer Grenzgebiete.**73**. Springer-Verlag. ISBN 3-540-06009-X. - Serre, Jean-Pierre (1973).
*A Course in Arithmetic*. Graduate Texts in Mathematics.**7**. Springer-Verlag. ISBN 0-387-90040-3. https://archive.org/details/courseinarithmet00serr. - Cassels, J.W.S. (1978).
*Rational Quadratic Forms*. London Mathematical Society Monographs.**13**. Academic Press. ISBN 0-12-163260-1.

Original source: https://en.wikipedia.org/wiki/ Meyer's theorem.
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