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On Euclidean ideal classes in certain Abelian extensions

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In this article, we show that certain abelian extensions K with unit rank greater than or equal to three have cyclic class group if and only if it has a Euclidean ideal class. This result improves an earlier result of Murty and Graves. One can improve this result up to unit rank 2 if one assumes the Elliott and Halberstam conjecture (see Conjecture 1 in preliminaries). These results are known under generalized Riemann hypothesis by the work of Lenstra (J Lond Math Soc 10:457–465) [see also Weinberger (Proc Symp Pure Math 24:321–332)].

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Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux UMR 5251, Université de Bordeaux, 351, Cours de la Libération, 33 405, Talence, France

    J.-M. Deshouillers

  2. Institute of Mathematical Sciences, HBNI, C.I.T Campus, Taramani, Chennai, 600 113, India

    S. Gun & J. Sivaraman

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  1. J.-M. Deshouillers
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  2. S. Gun
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  3. J. Sivaraman
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Correspondence to S. Gun.

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Deshouillers, JM., Gun, S. & Sivaraman, J. On Euclidean ideal classes in certain Abelian extensions. Math. Z. 296, 847–859 (2020). https://doi.org/10.1007/s00209-019-02434-2

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