Binary quadratic form: Difference between revisions

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There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms.<ref>{{harvnb|Weil|2001|loc=Ch.I §§VI, VIII}}</ref> The first problem concerning binary quadratic forms asks for the existence or construction of representations of integers by particular binary quadratic forms. The prime examples are the solution of [[Pell's equation]] and the representation of integers as [[sum of squares|sums of two squares]]. Pell's equation was already considered by the Indian mathematician [[Brahmagupta#Pell's equation|Brahmagupta]] in the 7th century CE. Several centuries later, his ideas were extended to a complete solution of Pell's equation known as the [[chakravala method]], attributed to either of the Indian mathematicians [[Jayadeva (mathematician)|Jayadeva]] or [[Bhāskara II]].<ref>{{harvnb|Weil|2001|loc=Ch.I §IX}}</ref> The problem of representing integers by sums of two squares was considered in the 6th century by [[Diophantus]].<ref>{{harvnb|Weil|2001|loc=Ch.I §IX}}</ref> In the 17th century, inspired while reading Diophantus's [[Arithmetica]], [[Fermat]] made several observations about representations by specific quadratic forms including that which is now know as [[Fermat's theorem on sums of two squares]].<ref>{{harvnb|Weil|2001|loc=Ch.II §§VIII-XI}}</ref> [[Euler]] provided the first proofs of Fermat's observations and added some new conjectures about representations by specific forms, without proof.<ref>{{harvnb|Weil|2001|loc=Ch.III §§VII-IX}}</ref>

The general theory of quadratic forms was initiated by [[Lagrange]] in 1775 in his ''[[List of important publications in mathematics#Recherches d'Arithmétique|Recherches d'Arithmétique]]''. Lagrange was the first to realize that "a coherent general theory required the simulatenous consideration of all forms."<ref>{{harvnb|Weil|2001|loc=p.318}}</ref> He was the first to recognize the importance of the discriminant and to define the essential notions of equivalence and reduction, which, according to Weil, have "dominated the whole subject of quadratic forms ever since".<ref>{{harvnb|Weil|2001|loc=p.317}}</ref> HeLagrange showed that there are finitely many equivalence classes of given discriminant, thereby defining for the first time an arithmetic [[Ideal class group|class number]]. His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of [[infrastructure (number theory)|infrastructure]]. In 1798, [[Adrien-Marie Legendre|Legendre]] published ''Essai sur la théorie des nombres'', which summarized the work of Euler and Lagrange and added some of his own contributions, including the first glimpse of a composition operation on forms.

The theory was vastly extended and refined by [[Carl Friedrich Gauss|Gauss]] in Section V of ''[[List of important publications in mathematics#Disquisitiones Arithmeticae|Disquisitiones Arithmeticae]]''. Gauss introduced a very general version of a composition operator that allows composing even forms of different discriminants and imprimitive forms. He replaced Lagrange's equivalence with the more precise notion of proper equivalence, and this enabled him to show that the primitive classes of given discriminant form a [[group (mathematics)|group]] under the composition operation. He introduced genus theory, which gives a powerful way to understand the quotient of the class group by the subgroup of squares. (Gauss and many subsequent authors wrote 2''b'' in place of ''b''; the modern convention allowing the coefficient of ''xy'' to be odd is due to [[Gotthold Eisenstein|Eisenstein]]).