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. 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 $p$ -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 (x 1, x 2, x 3, x 4) = x 21 + x 22 - px 23 - px 24

,

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 $p$ .

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. 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.

See also

References