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add material on primitive forms, fundamental discriminant and narrow class group of quadratic fields. |
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== Brief history ==
Binary quadratic forms were considered already by [[Fermat]], in particular, in the question of [[Fermat's theorem on sums of two squares|representations of numbers as sums of two squares]]. The theory of [[Pell's equation]] may be viewed as a part of the theory of binary quadratic forms. [[Lagrange]] in 1773 initiated the development of the general theory of quadratic forms. First systematic treatment of binary quadratic forms is due to [[Adrien-Marie Legendre|Legendre]]. Their theory was advanced much further by [[Carl Friedrich Gauss|Gauss]] in ''[[Disquisitiones Arithmeticae]]''. He considered questions of equivalence and reduction and introduced [[composition of binary quadratic forms]] (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 [[Ferdinand Eisenstein|Eisenstein]]). These investigations of Gauss strongly influenced both the arithmetical theory of quadratic forms in more than two variables and the subsequent development of algebraic number theory, where quadratic fields are replaced with more general [[number field]]s.
== Main questions ==
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