In
number theory, Meyer's theorem on
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s states that an
indefinite quadratic form
Indefinite may refer to:
* the opposite of definite in grammar
** indefinite article
** indefinite pronoun
* Indefinite integral, another name for the antiderivative
* Indefinite forms in algebra, see definite quadratic forms
* an indefinite matr ...
''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
The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent ''locally at all places'', i.e. equivalent over every completion o ...
(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''
12 + ''x''
22 − ''p''(''x''
32 + ''x''
42),
where ''p'' is a
prime number that is
congruent to 3 modulo 4. This can be proved by the method of
infinite descent In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold fo ...
using the fact that if the sum of two
perfect squares
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The usu ...
is divisible by such a ''p'' then each summand is divisible by ''p''.
See also
*
Lattice (group)
*
Oppenheim conjecture
In Diophantine approximation, the Oppenheim conjecture concerns representations of numbers by real quadratic forms in several variables. It was formulated in 1929 by Alexander Oppenheim and later the conjectured property was further strengthened by ...
References
*
*
*
* {{cite book , first=J.W.S. , last=Cassels , authorlink=J. W. S. Cassels , title=Rational Quadratic Forms , series=London Mathematical Society Monographs , volume=13 , publisher=
Academic Press , year=1978 , isbn=0-12-163260-1 , zbl=0395.10029
Quadratic forms
Theorems in number theory