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binary quadratic form
In mathematics, a binary quadratic form is a quadratic form in two variables. More concretely, it is a homogeneous polynomial of degree 2 in two variables
:
where ''a'', ''b'', ''c'' are the coefficients. Properties of binary quadratic forms depend in an essential way on the nature of the coefficients, which may be real numbers, rational numbers, or in the most delicate case, integers. Arithmetical aspects of the theory of binary quadratic forms are related to the arithmetic of quadratic fields and have been much studied, notably, by Gauss in Section V of ''Disquisitiones Arithmeticae''. The theory of binary quadratic forms has been extended in two directions: general number fields and quadratic forms in ''n'' variables.
==Brief history==
Binary quadratic forms were considered already by Fermat, in particular, in the question of 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 Legendre. Their theory was advanced much further by 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 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 fields.
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