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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Multiplicities of dihedral discriminants


Author: Daniel C. Mayer
Journal: Math. Comp. 58 (1992), 831-847, S55
MSC: Primary 11R29; Secondary 11R16, 11R20
MathSciNet review: 1122071
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Abstract: Given the discriminant $ {d_k}$ of a quadratic field k, the number of cyclic relative extensions $ N\vert 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1992-1122071-3
PII: S 0025-5718(1992)1122071-3
Keywords: Dihedral fields, quadratic ring class groups, cubic fields
Article copyright: © Copyright 1992 American Mathematical Society