it is a corollary of the biquadratic law in but did not provide a proof of either theorem. Dirichlet showed that the law in can be deduced from the law for without using quartic reciprocity. For an odd Gaussian prime and a Gaussian integer relatively prime to define the quadratic character for by: Let be distinct Gaussian primes where a and c are odd and b and d are even. Then Eisenstein integers Consider the following third root of unity: The ring of Eisenstein integers is For an Eisenstein prime and an Eisenstein integer with define the quadratic character for by the formula Let λ = a + bω and μ = c + dω be distinct Eisenstein primes where a and c are not divisible by 3 and b and d are divisible by 3. Eisenstein proved Imaginary quadratic fields The above laws are special cases of more general laws that hold for the ring of integers in any imaginary quadratic number field. Let k be an imaginary quadratic number field with ring of integers For a prime ideal with odd norm and define the quadratic character for as for an arbitrary ideal factored into prime ideals define and for define Let i.e. is an integral basis for For with odd norm define (ordinary) integers a, b, c, d by the equations, and a function If m = Nμ and n = Nν are both odd, Herglotz proved Also, if Then Polynomials over a finite field Let F be a finite field with q = pn elements, where p is an odd prime number and n is positive, and let F[x] be the ring of polynomials in one variable with coefficients in F. If and f is irreducible, monic, and has positive degree, define the quadratic character for F[x] in the usual manner: If is a product of monic irreducibles let Dedekind proved that if are monic and have positive degrees, Higher powers The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led 19th century mathematicians, including Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlet, Carl Gustav Jakob Jacobi, Gotthold Eisenstein, Richard Dedekind, Ernst Kummer, and David Hilbert to the study of general algebraic number fields and their rings of integers; specifically Kummer invented ideals in order to state and prove higher reciprocity laws. The ninth in the list of