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Bounds of the Ideal Class Numbers of Real Quadratic Function Fields

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Abstract

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields \( K = k{\left( {{\sqrt D }} \right)} \) over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P. R. China

    Kun Peng Wang

  2. State Key Laboratory of Information Security, Graduate School of Chinese Academy of Sciences, Beijing, 100039, P. R. China

    Kun Peng Wang

  3. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P. R. China

    Xian Ke Zhang

Authors
  1. Kun Peng Wang
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  2. Xian Ke Zhang
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Correspondence to Kun Peng Wang.

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Project supported by the NNSFC (No. 19771052)

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Wang, K.P., Zhang, X.K. Bounds of the Ideal Class Numbers of Real Quadratic Function Fields. Acta Math Sinica 20, 169–174 (2004). https://doi.org/10.1007/s10114-003-0284-0

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  • DOI: https://doi.org/10.1007/s10114-003-0284-0

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