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Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields by Integer Lattice Reduction

  • Conference paper
Number Theory

Abstract

Motivated by a constructive realization of generalized dihedral groups as Galois groups over Q and by Atkin’s primality test, we present an explicit construction of the Hilbert class fields (ring class fields) of imaginary quadratic fields (orders). This is done by first evaluating the singular moduli of level one for an imaginary quadratic order, and then constructing the “genuine” (i.e., level one) class equation. The equation thus obtained has integer coefficients of astronomical size, and this phenomenon leads us to the construction of the “reduced” class equations, i.e., the class equations of the singular moduli of higher levels. These, for certain levels, turn out to define the same Hilbert class field (ring class field) as the level one class equation, and to have coefficients of small size (e.g., seven digits). The construction of the “reduced” class equations was carried out on MACSYMA, using a refinement of the integer lattice reduction algorithm of Lenstra-Lenstra-Lavász, implemented on the Symbolics 3670 at Rensselaer Polytechnic Institute.

Erich Kaltofen was partially supported by the NSF Grant No. CCR-87–05363 and by an IBM Faculty Development Award. Noriko Yui was partially supported by the NSERC Grants No. A8566 and No. A9451, and by an Initiation Grant at Queen’s University.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Erich Kaltofen

  2. Department of Mathematics, Queen’s University, Kingston, Ontario, K7L3N6, Canada

    Noriko Yui

Authors
  1. Erich Kaltofen
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  2. Noriko Yui
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Columbia University, 10027, New York, NY, USA

    David V. Chudnovsky  & Gregory V. Chudnovsky  & 

  2. Department of Mathematics, City University of New York, City College, 10031, New York, NY, USA

    Harvey Cohn

  3. Lehman College, City University of New York, 10468, Bronx, NY, USA

    Melvyn B. Nathanson (Provost and Vice President of Academic Affairs) (Provost and Vice President of Academic Affairs)

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© 1991 Springer Science+Business Media New York

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Kaltofen, E., Yui, N. (1991). Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields by Integer Lattice Reduction. In: Chudnovsky, D.V., Chudnovsky, G.V., Cohn, H., Nathanson, M.B. (eds) Number Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4158-2_8

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  • DOI: https://doi.org/10.1007/978-1-4757-4158-2_8

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-97670-9

  • Online ISBN: 978-1-4757-4158-2

  • eBook Packages: Springer Book Archive

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