Key words and phrases: Binary quadratic forms, ideals, cycles of forms,  Buell, D. A., Binary Quadratic Forms, Clasical Theory and Modern Computations. “form” we mean an indefinite binary quadratic form with discriminant not a ..  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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The minimum absolute value represented by a class is zero for degenerate classes and positive for definite and indefinite classes.
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. If a form’s discriminant is a fundamental discriminantthen the form is primitive. There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms. Such a representation is a solution to the Pell equation described in the examples above.
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Binary Quadratic Forms
Articles lacking in-text citations from July All articles lacking in-text citations All articles with unsourced statements Articles with unsourced statements from March Binary quadratic forms are closely related to ideals in quadratic fields, this allows the class number of a quadratic field to be calculated by counting the number of reduced binary quadratic forms of a given discriminant.
His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of infrastructure.
Class groups have since become one of the central ideas in algebraic number theory. The equivalence relation above then arises from the general theory qudratic group actions.
Lehmer : Review: D. A. Buell, Binary quadratic forms, classical theory and applications
The oldest problem in the theory quadtatic binary quadratic forms is the representation problem: He introduced genus theory, which gives a powerful way to understand the quotient of the class group by the subgroup of squares. 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. For this reason, the former are called positive definite forms and the latter are negative definite.
This recursive description was discussed in Theon of Smyrna’s commentary on Euclid’s Elements.
Binary quadratic form
Each genus is the union of a finite number of equivalence classes of the same discriminant, with the number of classes depending only on the discriminant.
This page was last edited on 8 Novemberat We perform the following steps:. Gauss also considered a coarser notion of equivalence, with each coarse class called a genus of forms. Terminology has arisen for classifying classes and their forms in terms of their invariants.
In all, there are sixteen different solution pairs.
When the coefficients can be arbitrary complex numbersmost results are not specific to the case of two variables, so they are described in quadratic form. Even so, work on binary quadratic forms with integer coefficients continues to the present. Lagrange proved that for every value Dthere are only finitely many classes of binary quadratic forms with discriminant D.
A form is primitive if its content is 1, that is, if its coefficients are coprime. 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. Gauss introduced a very general version of a composition operator that allows composing even forms of different discriminants and imprimitive forms. 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.
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.
A quadratic form with integer coefficients is called an integral binary quadratic formoften abbreviated to binary quadratic form. Dirichlet published simplifications of the theory that made it accessible to a broader audience. Combined, the novelty and complexity made Section V notoriously difficult. Since the late nineteenth century, binary quadratic forms have given up their preeminence in algebraic number theory to quadratic and more general number fieldsbut advances specific to binary quadratic forms still occur on occasion.
There are only a finite number buwll pairs satisfying this constraint. It follows that equivalence defined this way is an equivalence relation and in particular that the forms in equivalent representations are equivalent forms.
We saw instances of this in the examples above: A third definition is a special case of the genus of a quadratic form in n variables.
It follows that the quadratic forms are partitioned into equivalence classes, called classes of quadratic forms. Their number is the class number of discriminant D. We present here Arndt’s method, because it remains rather general while being simple enough to be amenable to buelp by hand. The above equivalence conditions define an equivalence relation on the set of integral quadratic forms. Lagrange was the first to realize that “a coherent general theory required the simulatenous consideration of all forms.