Eisenstein Reciprocity

In order to prove higher reciprocity laws, the methods known to Gauss were soon found to be inadequate. The most obvious obstacle, namely the fact that the unique factorization theorem fails to hold for the rings ℤ[ζ_ℓ], was overcome by Kummer through the invention of his ideal numbers. The direct generalization of the proofs for cubic and quartic reciprocity, however, did not yield the general reciprocity theorem for ℓ-th powers: indeed, the most general reciprocity law that could be proved within the cyclotomic framework is Eisenstein’s reciprocity law. The key to its proof is the prime ideal factorization of Gauss sums; since we can express Gauss sums in terms of Jacobi sums and vice versa, the prime ideal factorization of Jacobi sums would do equally well.