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

 

The Euclidean algorithm for number fields and primitive roots


Authors: M. R. Murty and Kathleen L. Petersen
Journal: Proc. Amer. Math. Soc. 141 (2013), 181-190
MSC (2010): Primary 11A07, 11N36
Published electronically: May 25, 2012
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Abstract: Let $ K$ be a number field with unit rank at least four, containing a subfield $ M$ such that $ K/M$ is Galois of degree at least four. We show that the ring of integers of $ K$ is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann Hypothesis for Dedekind zeta functions. We prove this unconditionally.


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Additional Information

M. R. Murty
Affiliation: Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, University Avenue, Kingston, ON K7L 3N6, Canada
Email: murty@mast.queensu.ca

Kathleen L. Petersen
Affiliation: Department of Mathematics, Florida State University, 208 Love Building, Tallahassee, Florida 32306
Email: petersen@math.fsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11327-9
PII: S 0002-9939(2012)11327-9
Keywords: Primitive roots, Euclidean algorithm, large sieve
Received by editor(s): January 6, 2011
Received by editor(s) in revised form: June 22, 2011
Published electronically: May 25, 2012
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.