Editing Quadratic form (section)

== History ==
The study of quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is [[Fermat's theorem on sums of two squares]], which determines when an integer may be expressed in the form {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}}, where {{math|''x''}}, {{math|''y''}} are integers. This problem is related to the problem of finding [[Pythagorean triple]]s, which appeared in the second millennium BCE.<ref>[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_Pythagoras.html Babylonian Pythagoras]</ref>

In 628, the Indian mathematician [[Brahmagupta]] wrote ''[[Brāhmasphuṭasiddhānta]]'', which includes, among many other things, a study of equations of the form {{math|1=''x''<sup>2</sup> − ''ny''<sup>2</sup> = ''c''}}. He considered what is now called [[Pell's equation]], {{math|1=''x''<sup>2</sup> − ''ny''<sup>2</sup> = 1}}, and found a method for its solution.<ref>[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brahmagupta.html Brahmagupta biography]</ref> In Europe this problem was studied by [[William Brouncker, 2nd Viscount Brouncker|Brouncker]], [[Leonhard Euler|Euler]] and [[Joseph Louis Lagrange|Lagrange]].

In 1801 [[Carl Friedrich Gauss|Gauss]] published ''[[Disquisitiones Arithmeticae]],'' a major portion of which was devoted to a complete theory of [[binary quadratic form]]s over the [[integer]]s. Since then, the concept has been generalized, and the connections with [[quadratic number field]]s, the [[modular group]], and other areas of mathematics have been further elucidated.