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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From Wikipedia, the free encyclopedia. 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 saw instances of this in the examples above: This recursive description was discussed in Bue,l of Smyrna’s commentary on Euclid’s Elements.

Iterating this matrix action, we find that the infinite set of representations of 1 by f that were determined above are all equivalent. For this reason, the former are called positive definite forms and the latter are negative definite. If a form’s discriminant is a fundamental discriminantthen the form is primitive.

Retrieved from ” https: But the impact was not immediate. July Learn how and when to remove formz template message. 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. The equivalence relation above then arises from the general theory of group actions.


Thus, composition gives a well-defined function from pairs of binary quadratic forms to such classes.

We present here Arndt’s method, because it remains rather general while being simple enough to be amenable hinary computations by hand. We see that its first coefficient is well-defined, but the other two depend on the choice of B and C.

These investigations of Gauss strongly influenced both the arithmetical theory of quadratic forms in more than two variables and binarj subsequent development of algebraic number theory, where quadratic fields are replaced with more general number fields. Even so, work on binary quadratic forms with integer coefficients continues to the present. There is a closed formula [3].

In the first case, the sixteen representations were explicitly described. He described an algorithm, called reductionfor constructing a canonical representative in each class, the reduced formwhose coefficients ginary the smallest in a suitable sense.

Binary Quadratic Forms by Buell, Duncan a

Their number is the class number of discriminant D. Gauss gave a superior reduction algorithm in Disquisitiones Arithmeticae foorms, which has ever since the reduction algorithm most commonly given in textbooks. Pell’s equation was already considered by the Indian mathematician Brahmagupta in the 7th century CE.

Class groups have since become one of the central ideas in algebraic number theory. This article is about binary quadratic forms with integer coefficients. The notion of equivalence of forms can be extended to equivalent representations.


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.

Binary Quadratic Forms

Lagrange was the first to realize that “a coherent general theory required the simulatenous consideration of all forms. 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 mathematicsa binary quadratic form is a quadratic homogeneous polynomial in two variables. Another ancient problem involving quadratic forms asks us to solve Pell’s equation. Combined, the novelty and complexity made Section V notoriously difficult. The oldest problem in the theory of binary quadratic forms is the representation problem: In all, there are sixteen different solution pairs.

This page was last edited on 8 Novemberat 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. This article is entirely devoted to integral binary quadratic forms.