In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables
where a, b, c are the coefficients. When the coefficients can be arbitrary complex numbers, most 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 form, often 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 fields, but advances specific to binary quadratic forms still occur on occasion.
Pierre Fermat stated that if p is an odd prime then the equation has a solution iff , and he made similar statement about the equations , , and . and so on are quadratic forms, and the theory of quadratic forms gives a unified way of looking at and proving these theorems.
Another instance of quadratic forms is Pell's equation .
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.
The classical theta function of 2 variables is , if is a positive definite quadratic form then is a theta function.
Equivalence
Two forms f and g are called equivalent if there exist integers such that the following conditions hold:
For example, with and , , , and , we find that f is equivalent to , which simplifies to .
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 defined on equivalence classes of forms or a property shared by all forms in the same class.
Lagrange used a different notion of equivalence, in which the second condition is replaced by . 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 used occasionally below, when
has integer entries and determinant 1, the map is a (right) group action of on the set of binary quadratic forms. The equivalence relation above then arises from the general theory of group actions.
If , then important invariants include
The discriminant . The content, equal to the greatest common divisor of a, b, and c.
Terminology has arisen for classifying classes and their forms in terms of their invariants. A form of discriminant is definite if , degenerate if is a perfect square, and indefinite otherwise. A form is primitive if its content is 1, that is, if its coefficients are coprime. If a form's discriminant is a fundamental discriminant, then the form is primitive. Discriminants satisfy
Automorphisms
If f is a quadratic form, a matrix
in is an automorphism of f if . For example, the matrix
is an automorphism of the form . The automorphisms of a form form a subgroup of . When f is definite, the group is finite, and when f is indefinite, it is infinite and cyclic.
Representation
A binary quadratic form represents an integer if it is possible to find integers and satisfying the equation Such an equation is a representation of by .
Examples
Diophantus considered whether, for an odd integer , it is possible to find integers and for which . When , we have
so we find pairs that do the trick. We obtain more pairs that work by switching the values of and and/or by changing the sign of one or both of and . In all, there are sixteen different solution pairs. On the other hand, when , the equation
does not have integer solutions. To see why, we note that unless or . Thus, will exceed 3 unless is one of the nine pairs with and each equal to or 1. We can check these nine pairs directly to see that none of them satisfies , so the equation does not have integer solutions.
A similar argument shows that for each , the equation can have only a finite number of solutions since will exceed unless the absolute values and are both less than . There are only a finite number of pairs satisfying this constraint.