Given the binary quadratic form(1)with polynomial discriminant , let(2)(3)Then(4)where(5)(6)(7)so(8)Surprisingly, this is the same discriminant as before, but multiplied by the factor . The quantity is called the quadratic invariant modulus.
A set of algebraic invariants for a quantic such that any invariant of the quantic is expressible as a polynomial in members of the set. Gordan (1868) proved the existence of finite fundamental systems of algebraic invariants and covariants for any binary quadratic form, which in modern terminology would be stated that every binary quadratic form has a finite Hilbert basis. The complete systems of the quintic and sextic were also first obtained by Gordan in 1868.Hilbert (1890) subsequently proved the general Hilbertbasis theorem, which is a finiteness theorem for the related concept of syzygies.
A discriminant is a quantity (usually invariant under certain classes of transformations) which characterizes certain properties of a quantity's roots. The concept of the discriminant is used for binary quadratic forms, elliptic curves, metrics, modules, polynomials, quadratic curves, quadratic fields, quadratic forms, and in the second derivative test.