Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Some generalized Euclidean and $ 2$-stage Euclidean number fields that are not norm-Euclidean

Author: Jean-Paul Cerri
Journal: Math. Comp. 80 (2011), 2289-2298
MSC (2010): Primary 11Y40; Secondary 11R04, 12J15, 13F07
Published electronically: February 4, 2011
Supplement: Table supplement to this article.
MathSciNet review: 2813361
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. PARI/GP, version 2.1.3, Bordeaux, 2000,
  • 2. S. CAVALLAR, F. LEMMERMEYER, The Euclidean Algorithm in Cubic Number Fields, Proceedings Number Theory Eger 1996, (Györy, Pethö, Sos eds.), Gruyter 1998, 123-146. MR 1628838 (99e:11137)
  • 3. J.-P. CERRI, Spectres euclidiens et inhomogènes des corps de nombres, Thèse Université de Nancy 1 (2005).
  • 4. J.-P. 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 (2007f:11122)
  • 5. J.-P. CERRI, Euclidean minima of totally real number fields: Algorithmic determination, Mathematics of Computation 76 (2007), 1547-1575. MR 2299788 (2008d:11143)
  • 6. J.-P. CERRI, Tables of $ 2$-stage Euclidean number fields that are not norm-Euclidean,
  • 7. G.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 0406973 (53:10758a)
  • 8. G.E. COOKE, P.J. WEINBERGER, On the construction of Division Chains in Algebraic Number Rings, with Applications to $ SL_2$, Commun. Algebra 3 (1975), 481-524. MR 0387251 (52:8094)
  • 9. D.H. JOHNSON, C.S. QUEEN, A.N. SEVILLA, Euclidean quadratic number fields, Arch. Math. 44 (1985), 340-347. MR 788948 (87g:11137)
  • 10. F. LEMMERMEYER, The Euclidean algorithm in algebraic number fields, updated version of the article published in Expo. Math. 13 (1995), 385-416, available from MR 1362867 (96i:11115)
  • 11. H. POLLARD, The Theory of Algebraic Numbers, Math. Association of America, New York (1950). MR 0037319 (12:243g)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11Y40, 11R04, 12J15, 13F07

Retrieve articles in all journals with MSC (2010): 11Y40, 11R04, 12J15, 13F07

Additional Information

Jean-Paul Cerri
Affiliation: IMB, 351, cours de la Libération, 33400 Talence, France

Received by editor(s): September 11, 2009
Received by editor(s) in revised form: July 21, 2010
Published electronically: February 4, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society