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

$q(x,y)=ax^2+bxy+cy^2, \,$

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 2b 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.

Main questions

A classical question in the theory of integral quadratic forms (those with integer coefficients) is the representation problem: describe the set of numbers represented by a given quadratic form q. If the number of representations is finite then a further question is to give a closed formula for this number. The notion of equivalence of quadratic forms and the related reduction theory are the principal tools in addressing these questions.

Two integral forms are called equivalent if there exists an invertible integral linear change of variables that transforms the first form into the second. This defines an equivalence relation on the set of integral quadratic forms, whose elements are called classes of quadratic forms. Equivalent forms necessarily have the same discriminant

$D(f)=b^2-4ac, \quad D(f)\equiv 0,1 \pmod 4.$

Gauss proved that for every value D, there are only finitely many classes of binary quadratic forms with discriminant D. Their number is the class number of discriminant D. He described an algorithm, called reduction, for constructing a canonical representative in each class, the reduced form, whose coefficients are the smallest in a suitable sense. One of the deepest discoveries of Gauss was the existence of a natural composition law on the set of classes of binary quadratic forms of given discriminant, which makes this set into a finite abelian group called the class group of discriminant D. Gauss also considered a coarser notion of equivalence, under which the set of binary quadratic forms of a fixed discriminant splits into several genera of forms and each genus consists of finitely many classes of forms.

An integral binary quadratic form is called primitive if a, b, and c have no common factor. If a form's discriminant is a fundamental discriminant, then the form is primitive.^[1]

From a modern perspective, the class group of a fundamental discriminant D is isomorphic to the narrow class group of the quadratic field $\mathbf{Q}(\sqrt{D})$ of discriminant D.^[2] For negative D, the narrow class group is the same as the ideal class group, but for positive D it may be twice as big.

See also

• Bhargava cube
• Legendre symbol

Notes

1. Cohen 1993, §5.2
2. Fröhlich & Taylor 1993, Theorem 58

References

• Johannes Buchmann, Ulrich Vollmer: Binary Quadratic Forms, Springer, Berlin 2007, ISBN 3-540-46367-4
• Duncan A. Buell: Binary Quadratic Forms, Springer, New York 1989
• Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Berlin, New York: Springer-Verlag, ISBN 978-3-540-55640-4, MR 1228206
• Fröhlich, Albrecht; Taylor, Martin (1993), Algebraic number theory, Cambridge Studies in Advanced Mathematics 27, Cambridge University Press, ISBN 978-0-521-43834-6, MR 1215934

External links

• Peter Luschny, Positive numbers represented by a binary quadratic form
• A.V.Malyshev (2001), "Binary quadratic form", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4