|
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 (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 (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 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 == * Lattice (group) * Oppenheim conjecture 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Meyer's theorem」の詳細全文を読む スポンサード リンク
|