Binary quadratic form

Summary

A binary quadratic form is a quadratic form in two variables, $ax^2 + bxy + cy^2$, usually with integer coefficients. Its discriminant $b^2 - 4ac$ is unchanged by integral changes of variables of determinant $1$. Gauss developed the theory of these forms: which integers they represent, how they fall into equivalence classes, and how the classes combine into a group. This theory is equivalent to the arithmetic of ideals in quadratic fields, and the number of classes is a class number.