Meyer's theorem

In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form Template:Mvar in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation Template:Math has a non-zero real solution, then it has a non-zero rational solution. By clearing the denominators, an integral solution Template:Mvar 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 Template:Math of [[p-adic number|Template:Mvar-adic numbers]] for all Template:Mvar.

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

Template:Math,

where Template:Mvar 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 Template:Mvar, then each summand is divisible by Template:Mvar.

See also

• Lattice (group)
• Oppenheim conjecture