Reciprocity Laws: From Euler to Eisenstein
This book covers the development of reciprocity laws, starting from conjectures of Euler and discussing the contributions of Legendre, Gauss, Dirichlet, Jacobi, and Eisenstein. Readers knowledgeable in basic algebraic number theory and Galois theory will find detailed discussions of the reciprocity laws for quadratic, cubic, quartic, sextic and octic residues, rational reciprocity laws, and Eisensteins reciprocity law. An extensive bibliography will be of interest to readers interested in the history of reciprocity laws or in the current research in this area.
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abelian extensions Amer Artin assume automorphism biquadratic claim class group class number complete compute congruence conjecture Corollary cubic reciprocity law cubic residues cyclic cyclotomic fields Dedekind deduce defined denote derive Dirichlet divisor elliptic curves elliptic functions equation Euler Exercise fact factorization Fermat finite fields formula Gauss's Lemma Hasse hence Hilbert homomorphism ideal class implies integers Jacobi sums Jacobi symbol Jacobsthal sums K.S. Williams Kronecker Kummer Legendre Legendre's Lehmer Math mod q modp modulo Moreover n-th norm number theory octic reciprocity law odd prime polynomial primality tests primary prime prime ideal primes q Proc proof Proposition prove Q(Cm Q(CP quadratic Gauss sum quadratic number field quadratic reciprocity law quadratic residue quartic reciprocity law ramified rational Reine Angew residue modulo result root of unity splits square Stickelberger subfield supplementary laws Theorem Uber unramified Werke