On Euclidean ideal classes in certain Abelian extensions
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Mathematische Zeitschrift
On Euclidean ideal classes in certain Abelian extensions J.-M. Deshouillers1 · S. Gun2 · J. Sivaraman2 Received: 27 April 2018 / Accepted: 12 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract 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)]. Keywords Euclidean ideal classes · Galois Theory · Hilbert class fields · Brun’s Sieve · Bombieri–Vinogradov theorem · Linear Sieve Mathematics Subject Classification 11A05 · 13F07 · 11R04 · 11R27 · 11R32 · 11R37 · 11R42 · 11N36
1 Introduction Throughout the paper, let K be a number field with ring of integers O K and an infinite unit × group O× K . We shall call the rank of O K , the unit rank of K . Also let E K be the set of all fractional ideals of K containing O K . An ideal class [I] of Cl K is called a Euclidean ideal class if there exists a map ψ : E K → N such that for any ideal a ∈ [I] and for all ideals b ∈ E K and for all x ∈ ab\a, there exists z ∈ x + a such that ψ(z −1 ab) < ψ(b).
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S. Gun [email protected] J.-M. Deshouillers [email protected] J. Sivaraman [email protected]
1
Institut de Mathématiques de Bordeaux UMR 5251, Université de Bordeaux, 351, Cours de la Libération, 33 405 Talence, France
2
Institute of Mathematical Sciences, HBNI, C.I.T Campus, Taramani, Chennai 600 113, India
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J.-M. Deshouillers et al.
When O K is a PID, the principal ideal class is Euclidean if and only if O K is a Euclidean domain. Lenstra [17] introduced the notion of Euclidean ideal classes in order to generalize the concept of Euclidean domains and also to capture cyclic class groups. In the same paper, Lenstra showed that class group Cl K of K is cyclic if and only if it has a Euclidean ideal class, provided generalized Riemann hypothesis holds. When O K is a PID, this result was already proved by Weinberger [21]. Prior to this work, Motzkin [18], while trying to answer a question of Zariski, showed that if the unit rank of K is zero and O K is a PID then it does not necessarily mean that O K is Euclidean. In the same paper, Motzkin also devised a criterion to determine when an integral domain is Euclidean. Using this criterion and its variant deduced by Clark and Murty [3], several authors [10,12,13,16,19] have tried to prove the result of Weinberger unconditionally. In a recent work, Graves [7] generalized Clark and Murty’s criterion in the set-up of Euclidean ideal classes. Using this criterion, Graves and Murty [8] tried to remove generalized Riemann hypothesis from the work of Lenstra for ce
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