Some generalized Euclidean and $2$-stage Euclidean number fields that are not norm-Euclidean
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- by Jean-Paul Cerri PDF
- Math. Comp. 80 (2011), 2289-2298 Request permission
Abstract:
We give examples of Generalized Euclidean but not norm-Euclidean number fields of degree greater than 2. In the same way we give examples of 2-stage Euclidean but not norm-Euclidean number fields of degree greater than 2. In both cases, no such examples were known.References
- Pari/GP, version 2.1.3, Bordeaux, 2000, http://pari.math.u-bordeaux.fr
- Stefania Cavallar and Franz Lemmermeyer, The Euclidean algorithm in cubic number fields, Number theory (Eger, 1996) de Gruyter, Berlin, 1998, pp. 123–146. MR 1628838, DOI 10.1023/A:1008244007194
- J.-P. Cerri, Spectres euclidiens et inhomogènes des corps de nombres, Thèse Université de Nancy 1 (2005).
- Jean-Paul Cerri, Inhomogeneous and Euclidean spectra of number fields with unit rank strictly greater than 1, J. Reine Angew. Math. 592 (2006), 49–62. MR 2222729, DOI 10.1515/CRELLE.2006.022
- Jean-Paul Cerri, Euclidean minima of totally real number fields: algorithmic determination, Math. Comp. 76 (2007), no. 259, 1547–1575. MR 2299788, DOI 10.1090/S0025-5718-07-01932-1
- J.-P. Cerri, Tables of $2$-stage Euclidean number fields that are not norm-Euclidean, http://www.math.u-bordeaux1.fr/~cerri/publications.html
- George E. Cooke, A weakening of the Euclidean property for integral domains and applications to algebraic number theory. I, J. Reine Angew. Math. 282 (1976), 133–156. MR 406973, DOI 10.1515/crll.1976.282.133
- George Cooke and Peter J. Weinberger, On the construction of division chains in algebraic number rings, with applications to $\textrm {SL}_{2}$, Comm. Algebra 3 (1975), 481–524. MR 387251, DOI 10.1080/00927877508822057
- David H. Johnson, Clifford S. Queen, and Alicia N. Sevilla, Euclidean real quadratic number fields, Arch. Math. (Basel) 44 (1985), no. 4, 340–347. MR 788948, DOI 10.1007/BF01235777
- Franz Lemmermeyer, The Euclidean algorithm in algebraic number fields, Exposition. Math. 13 (1995), no. 5, 385–416. MR 1362867
- Harry Pollard, The Theory of Algebraic Numbers, Carus Monograph Series, no. 9, Mathematical Association of America, Buffalo, N.Y., 1950. MR 0037319
Additional Information
- Jean-Paul Cerri
- Affiliation: IMB, 351, cours de la Libération, 33400 Talence, France
- Email: Jean-Paul.Cerri@math.u-bordeaux1.fr
- Received by editor(s): September 11, 2009
- Received by editor(s) in revised form: July 21, 2010
- Published electronically: February 4, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 80 (2011), 2289-2298
- MSC (2010): Primary 11Y40; Secondary 11R04, 12J15, 13F07
- DOI: https://doi.org/10.1090/S0025-5718-2011-02468-3
- MathSciNet review: 2813361