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. 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 ℚp of p-adic numbers for all p.
Meyer's theorem is the 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(x1,x2,x3,x4) = x2
1 + x2
2 − px2
3 − px2
4,
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
editReferences
edit- 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. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.
- Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. Vol. 7. Springer-Verlag. ISBN 0-387-90040-3. Zbl 0256.12001.
- Cassels, J.W.S. (1978). Rational Quadratic Forms. London Mathematical Society Monographs. Vol. 13. Academic Press. ISBN 0-12-163260-1. Zbl 0395.10029.