Meyer's theorem

{{Short description|Indefinite quadratic forms in > 4 variables over the rationals nontrivially represent 0}}

{{For|the theorem in computational complexity theory|P/poly}}

In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form {{Mvar|Q}} in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation {{Math|1=Q(x) = 0}} has a non-zero real solution, then it has a non-zero rational solution. By clearing the denominators, an integral solution {{Mvar|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 {{Math|ℚ_p}} of p-adic number for all {{Mvar|p}}.

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

:{{Math|1=Q(x₁,x₂,x₃,x₄) = x{{su|p=2|b=1}} + x{{su|p=2|b=2}} − px{{su|p=2|b=3}} − px{{su|p=2|b=4}}}},

where {{Mvar|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 {{Mvar|p}}, then each summand is divisible by {{Mvar|p}}.