A family of number fields with unit rank at least $4$ that has Euclidean ideals
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- by Hester Graves and M. Ram Murty PDF
- Proc. Amer. Math. Soc. 141 (2013), 2979-2990 Request permission
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.References
- Alina Carmen Cojocaru and M. Ram Murty, An introduction to sieve methods and their applications, London Mathematical Society Student Texts, vol. 66, Cambridge University Press, Cambridge, 2006. MR 2200366
- Hester Graves and Nick Ramsey, Euclidean ideals in quadratic imaginary fields, J. Ramanujan Math. Soc. 26 (2011), no. 1, 85–97. MR 2789745
- Hester Graves, Growth Results and Euclidean Ideals, submitted, arXiv:1008.2479.
- Hester Graves, $\Bbb Q(\sqrt {2},\sqrt {35})$ has a non-principal Euclidean ideal, Int. J. Number Theory 7 (2011), no. 8, 2269–2271. MR 2873154, DOI 10.1142/S1793042111004988
- Rajiv Gupta and M. Ram Murty, A remark on Artin’s conjecture, Invent. Math. 78 (1984), no. 1, 127–130. MR 762358, DOI 10.1007/BF01388719
- H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
- Malcolm Harper, $\Bbb Z[\sqrt {14}]$ is Euclidean, Canad. J. Math. 56 (2004), no. 1, 55–70. MR 2031122, DOI 10.4153/CJM-2004-003-9
- Malcolm Harper and M. Ram Murty, Euclidean rings of algebraic integers, Canad. J. Math. 56 (2004), no. 1, 71–76. MR 2031123, DOI 10.4153/CJM-2004-004-5
- H. W. Lenstra Jr., Euclidean ideal classes, Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978) Astérisque, vol. 61, Soc. Math. France, Paris, 1979, pp. 121–131. MR 556669
- M. Ram Murty, Problems in analytic number theory, Graduate Texts in Mathematics, vol. 206, Springer-Verlag, New York, 2001. Readings in Mathematics. MR 1803093, DOI 10.1007/978-1-4757-3441-6
- William Stein, SAGE Mathematics Software (version 4.4.4), The SAGE Group, 2010, http://www.sagemath.org/.
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
- MR Author ID: 128555
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 2979-2990
- MSC (2010): Primary 11-XX, 13F07
- DOI: https://doi.org/10.1090/S0002-9939-2013-11602-3
- MathSciNet review: 3068950