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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A family of number fields with unit rank at least $ 4$ that has Euclidean ideals


Authors: Hester Graves and M. Ram Murty
Journal: Proc. Amer. Math. Soc. 141 (2013), 2979-2990
MSC (2010): Primary 11-XX, 13F07
Published electronically: May 10, 2013
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Abstract: We will prove that if the unit rank of a number field with cyclic class group is large enough and if the Galois group of its Hilbert class field over $ \mathbb{Q}$ is abelian, then every generator of its class group is a Euclidean ideal class. We use this to prove the existence of a non-principal Euclidean ideal class that is not norm-Euclidean by showing that $ \mathbb{Q}(\sqrt {5}, \sqrt {21}, \sqrt {22})$ has such an ideal class.


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Additional Information

Hester Graves
Affiliation: Department of Mathematics, Queen’s University, 99 University Avenue, Kingston, Ontario, K7L 3N6, Canada
Email: gravesh@mast.queensu.ca

M. Ram Murty
Affiliation: Department of Mathematics, Queen’s University, 99 University Avenue, Kingston, Ontario, K7L 3N6, Canada

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11602-3
PII: S 0002-9939(2013)11602-3
Received by editor(s): May 27, 2011
Received by editor(s) in revised form: October 5, 2011, and November 16, 2011
Published electronically: May 10, 2013
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.