Binary quadratic form

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.

Two forms f and g are called equivalent if there exist integers ${\displaystyle \alpha ,\beta ,\gamma ,{\text{ and }}\delta }$ such that the following conditions hold:

For example, with ${\displaystyle f=x^{2}+4xy+2y^{2}}$ and ${\displaystyle \alpha =-3}$, ${\displaystyle \beta =2}$, ${\displaystyle \gamma =1}$, and ${\displaystyle \delta =-1}$, we find that f is equivalent to ${\displaystyle g=(-3x+2y)^{2}+4(-3x+2y)(x-y)+2(x-y)^{2}}$, which simplifies to ${\displaystyle -x^{2}+4xy-2y^{2}}$.

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