Key words and phrases: Binary quadratic forms, ideals, cycles of forms, [2] Buell, D. A., Binary Quadratic Forms, Clasical Theory and Modern Computations. “form” we mean an indefinite binary quadratic form with discriminant not a .. [1] D. A. Buell, Binary quadratic forms: Classical theory and modern computations. Citation. Lehmer, D. H. Review: D. A. Buell, Binary quadratic forms, classical theory and applications. Bull. Amer. Math. Soc. (N.S.) 23 (), no. 2,

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Retrieved from ” https: If there is a need to distinguish, sometimes forms are called properly equivalent using the definition above and improperly equivalent if they are equivalent in Lagrange’s sense. There are only a formz number of pairs satisfying this constraint.

Binary Quadratic Forms

The composition operation on equivalence classes is defined by first defining composition of forms and then showing that this induces a well-defined operation on classes. Another ancient problem involving quadratic forms asks us to solve Pell’s equation. Gauss introduced a very general version of a composition operator that allows composing even forms of different discriminants and binay forms.

Please help to improve this article by introducing more precise fors. Section V of Disquisitiones contains truly revolutionary ideas and involves very complicated computations, sometimes left to the reader.

We will consider such operations in a separate section below. In matrix terminology, which is used occasionally below, when. The notion of equivalence of forms can be extended to equivalent representations.

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 under the composition operation. Such a representation is a solution to the Pell equation described in the examples above. For this reason, the former are called positive definite forms and the latter are negative definite. The word “roughly” indicates two caveats: This article is about binary quadratic forms with integer coefficients.


Binary quadratic form – Wikipedia

Alternatively, we may view the result of composition, not as a form, but as an equivalence class of forms modulo the action of the group of matrices of the form. 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.

We saw instances of this in the examples above: There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms. This choice is motivated by their status as the driving force behind the development of algebraic number theory. Combined, the novelty and complexity made Section V notoriously difficult. One way to make this a well-defined operation is to make an arbitrary convention for how to choose B —for instance, choose B to be the smallest positive solution to the system of congruences above.

This includes numerous results about quadratic number fields, which can often be translated into the language of binary quadratic forms, but also includes developments about forms themselves or that originated by thinking about forms, including Shanks’s infrastructure, Zagier’s reduction algorithm, Conway’s topographs, and Bhargava’s reinterpretation of composition through Bhargava cubes.

The number of representations of an integer n by a form f is finite if f is definite and infinite if f is indefinite. It follows that the quadratic forms are partitioned into equivalence classes, called classes of quadratic forms. Lagrange proved that for every value Dthere are only finitely many classes of binary quadratic forms with discriminant D. If a form’s discriminant is a fundamental discriminantthen the form is primitive. Changing signs of x and y in a solution gives another solution, so it is enough to seek just solutions in positive integers.


This operation is substantially more complicated [ citation needed ] than composition of forms, but arose first historically. But the impact was not immediate. The equivalence relation above then arises from the general theory of group actions. 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.

InZagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss’s. Supplement XI introduces ring theoryand from then on, especially after the publication of Hilbert’s Zahlberichtthe theory of binary quadratic forms lost its preeminent position in algebraic number theory and became overshadowed by the more general theory of algebraic number fields.

This article includes a list of referencesbut its sources remain unclear because it has insufficient inline citations. This article is entirely devoted to integral binary quadratic forms. We perform the following steps:. Gauss also considered a coarser notion of equivalence, with each coarse class called a genus of forms. It follows that equivalence defined this way is an equivalence relation and in particular that the forms in equivalent representations are equivalent forms.

A quadratic form with integer coefficients is called an integral binary quadratic formoften abbreviated to binary quadratic form.

A complete set of representatives for these classes can be given in terms of reduced forms defined in the section below.