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<p>Let <i>K</i> be a number field. A Euclidean ideal class is a class [<i>C</i>] in the ideal class group of <i>K</i> that admits a map <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\psi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> serving as a Euclidean algorithm for <i>C</i>. Lenstra (Astérisque 61:121–131, 1979) introduced this notion and showed that, assuming the generalized Riemann hypothesis, every non-imaginary quadratic number field has a Euclidean ideal class if and only if its class group is cyclic. In this paper, we extend a method of Graves (Int J Number Theory 7(8):2269–2271, 2011) to prove that certain real biquadratic fields whose class number is a power of two possess a non-principal Euclidean ideal class.</p>

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Non-principal Euclidean ideals in certain real biquadratic fields

  • 期刊论文

  • 发表日期：

    2026年4月13日

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  • Hamza Boubker,
  • Mohammed Taous

摘要

Let K be a number field. A Euclidean ideal class is a class [C] in the ideal class group of K that admits a map \(\psi \) ψ serving as a Euclidean algorithm for C. Lenstra (Astérisque 61:121–131, 1979) introduced this notion and showed that, assuming the generalized Riemann hypothesis, every non-imaginary quadratic number field has a Euclidean ideal class if and only if its class group is cyclic. In this paper, we extend a method of Graves (Int J Number Theory 7(8):2269–2271, 2011) to prove that certain real biquadratic fields whose class number is a power of two possess a non-principal Euclidean ideal class.