Multiplicities of Dihedral Discriminants

Given the discriminant dk of a quadratic field k, the number of cyclic relative extensions N\k of fixed odd prime degree p with dihedral absolute Galois group of order 2p , which share a common conductor /, is called the multiplicity of the dihedral discriminant d^ = f2^p~^d^ . In this paper, general formulas for multiplicities of dihedral discriminants are derived by analyzing the p-rank of the ring class group mod / of k . For the special case p = 3 , c4 = -3 , an elementary proof is given additionally. The theory is illustrated by a discussion of all known discriminants of multiplicity > 5 of totally real and complex cubic fields. Introduction Let p be an odd prime and AT|Q a cyclic extension of degree p. Then it is well known [9, 15, 7] that the conductor of K must have the form / = Pe 'Qi • ■ ■ It> where e = 0 or e = 2, /> 0, and the q¡ are pairwise distinct rational primes satisfying q¡ = X (mod p) for i = X, ... , t. The discriminant of K is just a power of the conductor, d¡( = fp~x . If, for any positive integer /, the number of cyclic extensions K\Q of degree p which share the same conductor / is denoted by m(f), then /-i/ p where p = dimF,(Qx(/)/Q* -qx(f)p) = dimr,(SyL, U(Z/fZ) ®Zp ¥p) = t + w with ®x(f) = {r£®x \(r,f) = X}, Q; = {reQx \r= X (mod*/)}, and w = \e. Moebius inversion yields an explicit formula for m(f): m(pe-qx---qt) = (p-l),+w-1 . It is the aim of the present paper to establish similar formulas for multiplicities of discriminants in the case of non-Galois extensions L|Q of degree p with dihedral normal closure yV of degree 2p. For the sake of illustration, the formulas are applied to discriminants with multiplicities up to 16 of non-Galois Received December 10, 1990; revised April 5, 1991. 1991 Mathematics Subject Classification. Primary 11R20, 11R11, 11R16.