Translator Disclaimer
May 2020 On Existence of Euclidean Ideal Classes in Real Cubic and Quadratic Fields with Cyclic Class Group
Sanoli Gun, Jyothsnaa Sivaraman
Michigan Math. J. 69(2): 429-448 (May 2020). DOI: 10.1307/mmj/1580180457

Abstract

Lenstra introduced the notion of Euclidean ideal classes for number fields to study cyclicity of their class groups. In particular, he showed that the class group of a number field with unit rank greater than or equal to one is cyclic if and only if it has a Euclidean ideal class. The only if part in the above result is conditional on the extended Riemann hypothesis. Graves and Murty showed that one does not require the extended Riemann hypothesis if the unit rank of the number field is greater than or equal to four and its Hilbert class field is abelian over rationals. In this article, we study real cubic and quadratic fields with cyclic class groups and show that they have a Euclidean ideal class under certain conditions.

Citation

Download Citation

Sanoli Gun. Jyothsnaa Sivaraman. "On Existence of Euclidean Ideal Classes in Real Cubic and Quadratic Fields with Cyclic Class Group." Michigan Math. J. 69 (2) 429 - 448, May 2020. https://doi.org/10.1307/mmj/1580180457

Information

Received: 7 June 2018; Revised: 23 October 2019; Published: May 2020
First available in Project Euclid: 28 January 2020

zbMATH: 07244379
MathSciNet: MR4104380
Digital Object Identifier: 10.1307/mmj/1580180457

Subjects:
Primary: 11A05 , 11N36 , 11R04 , 11R27 , 11R32 , 11R37 , 11R42 , 13F07

Rights: Copyright © 2020 The University of Michigan

JOURNAL ARTICLE
20 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.69 • No. 2 • May 2020
Back to Top