Multiplicities of dihedral discriminants

Mayer, Daniel C.

doi:10.1090/S0025-5718-1992-1122071-3

Multiplicities of dihedral discriminants
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Math. Comp. 58 (1992), 831-847 Request permission

Abstract:

Given the discriminant ${d_k}$ 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 f, is called the multiplicity of the dihedral discriminant ${d_N} = {f^{2(p - 1)}}d_k^p$. In this paper, general formulas for multiplicities of dihedral discriminants are derived by analyzing the p-rank of the ring class group mod f of k. For the special case $p = 3,{d_k} = - 3$, an elementary proof is given additionally. The theory is illustrated by a discussion of all known discriminants of multiplicity $\geq 5$ of totally real and complex cubic fields.

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