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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In mathematicsa binary quadratic form is a quadratic homogeneous polynomial in two variables. 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. A quadratic form with integer coefficients is called an integral binary quadratic formoften abbreviated to binary quadratic form. This article is entirely devoted to integral binary quadratic forms.
This choice is motivated by their status as the driving force behind the development of algebraic number theory. 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. Binary quadratic forms are closely related to ideals in quadratic fields, this allows the class number of a quadratic field to be quadragic by counting the number of reduced binary quadratic forms of a given discriminant.
The above equivalence conditions define an equivalence relation on the set of integral quadratic forms. It follows that the quadratic forms are partitioned into equivalence classes, called classes of quadratic forms. A class invariant can mean either a function bue,l on equivalence classes of forms or a property shared by all forms in the same class. Since Gauss it has been recognized that this definition is inferior to that given above.
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. In matrix terminology, which is binarry occasionally below, when. The equivalence relation above then arises from the general theory of group actions.
Terminology has arisen for classifying classes and their forms in terms of their invariants. A form is primitive if its content is 1, that is, if its coefficients are coprime. If a form’s discriminant is a fundamental discriminantthen the form is primitive. For example, the matrix.
When f is definite, the group is finite, and when f is indefinite, it is infinite and cyclic. In all, there are sixteen different solution pairs.
There are only a finite number of pairs satisfying this constraint. Another ancient problem involving quadratic forms asks us to solve Pell’s equation.
Changing signs of x and y in a solution gives another solution, so it is enough to seek just solutions in positive integers. This recursive description was discussed in Theon of Smyrna’s commentary on Euclid’s Elements. The oldest problem in the theory of binary quadratic forms is the representation problem: In the first case, the sixteen representations were explicitly described.
There is a closed formula . The minimum absolute value represented by a class is zero for degenerate classes and positive for definite and indefinite classes.
Binary quadratic form
For this reason, the former are called positive definite forms and the latter are negative definite. The number of representations of an integer n by a form f is finite if f is definite and infinite if f is indefinite. We saw instances of this in the examples above: The notion of equivalence of forms can be extended to equivalent representations.
It follows that equivalence defined this way is an equivalence relation and in particular that the forms in equivalent representations are equivalent forms. Such a representation is a solution to the Pell equation described in the examples above. Iterating this matrix action, we find that the infinite set of representations of 1 by f that were determined above are all equivalent. A complete set of representatives for these classes can be given in terms of reduced forms defined in the section below.
Lagrange proved that for every value Dthere 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 reductionfor constructing a canonical representative in each class, the reduced formwhose coefficients are the smallest in a suitable sense. Gauss gave a superior reduction algorithm in Disquisitiones Arithmeticaewhich has ever since the reduction algorithm most commonly given in textbooks.
InZagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss’s. Class groups have since become one of the central ideas in algebraic number theory. The word “roughly” indicates two caveats: 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. This operation is substantially more complicated [ citation needed ] than composition of forms, but arose first historically.
We will consider such operations in a separate section below. A variety of definitions of composition of forms has been given, often in an attempt to simplify the extremely technical and general definition of Gauss. We present here Arndt’s method, because it remains rather general while being simple enough to be amenable to computations by hand. An alternative definition is described at Bhargava cubes.
We perform the following steps:. We see that its first coefficient is well-defined, but the other two depend on the choice of B and C.
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.
Thus, composition gives a well-defined function from pairs of binary quadratic forms to such classes. Gauss also considered a coarser notion of equivalence, with each coarse class called a genus of forms. 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.
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.
A third definition is a special case of the genus of a quadratic form in n variables. This states that forms are in the same genus if they are locally equivalent at all rational primes including the Archimedean place.
There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms. The prime examples are the solution of Pell’s equation and the representation of integers as sums of two squares.
Pell’s equation was already considered by the Indian mathematician Brahmagupta in the 7th century CE. Lagrange was the first to realize that “a coherent general theory required the simulatenous consideration of all forms. His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of infrastructure. Gauss introduced a very general version of a composition operator that allows composing even forms of different discriminants and imprimitive forms.
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. He introduced genus theory, which gives a powerful way to understand the quotient of the class group by the subgroup of squares. 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. But the impact was not immediate.
Section V of Disquisitiones contains truly revolutionary ideas quadratkc involves very complicated computations, sometimes left to the reader.
Combined, the novelty and complexity made Section V notoriously difficult. Dirichlet published simplifications of the theory that made it accessible to a broader audience.
Jagy , Kaplansky : Indefinite binary quadratic forms with Markov ratio exceeding 9
The third edition of this work includes two supplements by Dedekind. 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. Even so, work on binary quadratic forms with integer coefficients continues to the present. 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.
From Wikipedia, the free encyclopedia. This article is about binary quadratic forms with integer coefficients.
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