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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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. We perform the following steps:. 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.
In the first case, the sixteen representations were explicitly described. 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.
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. 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.
In mathematicsa binary quadratic form is a quadratic homogeneous polynomial in two variables.
Binary quadratic form
The number of representations of an integer n by a form f is finite if f is definite and infinite if f is indefinite. This article is entirely devoted to integral binary quadratic forms. Quadrxtic 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.
Retrieved from ” https: Since Gauss it has been recognized that this definition is inferior to that given above. A complete set of representatives for these classes can be given in terms of reduced forms defined in the section below.
July Learn how and when to remove this template message. This recursive description was discussed in Theon of Smyrna’s commentary on Euclid’s Elements. Another ancient problem involving quadratic forms asks us to solve Pell’s equation.
The equivalence relation above then arises from the general theory of group actions. Since the late nineteenth century, binary quadratic forms have given up their preeminence in algebraic number theory to forrms and more general number fieldsbut advances specific to binary quadratic forms still occur on occasion.
For example, the matrix.
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. The notion of equivalence of forms can be extended to equivalent representations. Changing signs of x and y in a solution gives another solution, so it is enough to seek just solutions in positive integers. In all, there are sixteen different solution pairs.
But the impact was not immediate. Iterating this matrix action, we find that the infinite set of representations of 1 by f that were determined above are all equivalent. When f is definite, the group is finite, and when f is indefinite, it is infinite and cyclic. A form is primitive if its content is 1, that is, if its coefficients are coprime.
Binary Quadratic Forms
Their number is the class number of discriminant D. For this reason, the former are called positive definite forms and the latter are negative definite. It follows that equivalence defined this way is an equivalence relation and in particular that the forms in equivalent representations are equivalent forms.
If quadrxtic form’s discriminant is a fundamental discriminantthen the form is primitive.
Lagrange was the first to realize that “a coherent general theory required the simulatenous consideration of all forms. Gauss gave a superior reduction algorithm in Disquisitiones Arithmeticaewhich has ever since the reduction algorithm most commonly given in textbooks. Combined, the novelty and complexity made Section V notoriously difficult.
Binary quadratic form – Wikipedia
InZagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss’s. An alternative definition is described at Bhargava cubes. The third edition of buel work includes two supplements by Dedekind.
Lagrange proved that for every value Dthere are only finitely many classes of binary quadratic forms with 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. Dirichlet published simplifications of the theory that made it accessible to a broader audience. This article includes a list of referencesbut its sources remain unclear because it has insufficient inline citations.
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