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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Multiplicities of dihedral discriminants
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by Daniel C. Mayer PDF
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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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Math. Comp. 58 (1992), 831-847
  • MSC: Primary 11R29; Secondary 11R16, 11R20
  • DOI: https://doi.org/10.1090/S0025-5718-1992-1122071-3
  • MathSciNet review: 1122071