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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Each genus is the union of a finite form of equivalence classes of the same discriminant, with the number of classes depending only on the discriminant.

Gauss gave a superior reduction algorithm in Disquisitiones Arithmeticaewhich has ever since the reduction algorithm most commonly given in textbooks. He described an algorithm, called reductionfor constructing a canonical representative in each class, the reduced formwhose coefficients are the smallest in a suitable sense. ibnary

Binary Quadratic Forms

There is a closed formula [3]. 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.

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. He introduced genus theory, which gives a powerful way to understand the quotient of the class group by the subgroup of squares.


His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of infrastructure.

From Wikipedia, the free encyclopedia. Gauss also considered a coarser notion of equivalence, with each coarse class called a genus of forms. Quadratuc equivalence relation above then arises from the general theory of group actions.

Binary Quadratic Forms by Buell, Duncan a

In mathematicsa binary quadratic form is a quadratic homogeneous polynomial in two variables. This article is about binary quadratic forms with integer coefficients. A class invariant can mean either a function defined on equivalence classes of forms or a property shared by all forms in the same class. For example, the matrix. If a form’s discriminant is a fundamental discriminantthen the form is primitive.

Combined, the quzdratic and complexity made Section V notoriously difficult. This article is entirely devoted to integral binary quadratic forms. It follows that the quadratic forms are partitioned into equivalence classes, called classes of quadratic forms. An alternative definition is described at Bhargava cubes. Section V of Disquisitiones contains truly revolutionary ideas and involves very complicated computations, sometimes left to the reader.

Jagy , Kaplansky : Indefinite binary quadratic forms with Markov ratio exceeding 9

Class groups have since become one of the central ideas in algebraic number theory. The word “roughly” indicates two caveats: In the context of binary quadratic forms, genera can be defined either through congruence classes of numbers represented by forms or by genus characters defined on the set of forms.

Views Read Edit View history. We present here Arndt’s method, because it remains rather general while being simple enough to be amenable to computations by hand. Their number is the class number of discriminant D.


Lagrange was the first to realize that “a coherent general theory required the simulatenous consideration of all forms. A quadratic form with integer coefficients is called an integral binary quadratic formoften abbreviated to binary quadratic form. 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.

July Learn how and when to remove this template message. The above equivalence conditions define an equivalence relation on the set of integral quadratic forms. But the impact was not immediate. Lagrange proved that for every value Dthere are only finitely many classes of binary quadratic forms with discriminant D.

Binary quadratic form

The minimum absolute value represented by a class is zero for degenerate classes bueell positive for definite and indefinite classes.

Such a representation is a solution to the Pell equation described in the examples above. Please help to improve this article by introducing more precise citations. A third definition is a special case of the genus of a quadratic form in n variables. Terminology has arisen for classifying classes and their forms in terms of their invariants.