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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
Posted:
May 25, 2012
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Additional Information
Abstract: Let  be a number field with unit rank at least four, containing a subfield  such that  is Galois of degree at least four. We show that the ring of integers of  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.
- 1.
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Carmen Cojocaru and M.
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Ram Murty, and V.
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(2008j:11001)
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M. Ram Murty and Kathleen L. Petersen, A Bombieri-Vinogradov theorem for all number fields, Trans. Amer. Math. Soc., to appear.
- 11.
M.
Ram Murty and Kathleen
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orbifolds, Groups and symmetries, CRM Proc. Lecture Notes,
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𝑃𝑆𝐿₂(𝒪_{𝒦}), Proc. Amer. Math. Soc. 136 (2008), no. 7, 2387–2393. MR 2390505
(2008m:11091), http://dx.doi.org/10.1090/S0002-9939-08-09262-9
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Samuel, About Euclidean rings, J. Algebra 19
(1971), 282–301. MR 0280470
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(49 #2671)
- 1.
- David A. Clark and M. Ram Murty, The Euclidean algorithm for Galois extensions of
 ,
J. Reine Angew. Math. 459 (1995), 151-162. MR 1319520 (96h:11104)
- 2.
- 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 (2006k:11184)
- 3.
- Rajiv Gupta and M. Ram Murty, A remark on Artin's conjecture, Invent. Math. 78 (1984), no. 1, 127-130. MR 762358 (86d:11003)
- 4.
- Rajiv Gupta, M. Ram Murty, and V. Kumar Murty, The Euclidean algorithm for
 -integers, Number theory (Montreal, Que., 1985), CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 189-201. MR 894323 (88h:11088)
- 5.
- Malcolm Harper,
![$ \mathbb{Z}[\sqrt {14}]$](images/img7.gif) is Euclidean, Canad. J. Math. 56 (2004), no. 1, 55-70. MR 2031122 (2005f:11236)
- 6.
- Malcolm Harper and M. Ram Murty, Euclidean rings of algebraic integers, Canad. J. Math. 56 (2004), no. 1, 71-76. MR 2031123 (2005h:11261)
- 7.
- D. R. Heath-Brown, Artin's conjecture for primitive roots, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 145, 27-38. MR 830627 (88a:11004)
- 8.
- Henryk Iwaniec, Rosser's sieve, Acta Arith. 36 (1980), no. 2, 171-202. MR 581917 (81m:10086)
- 9.
- M. Ram Murty, Problems in analytic number theory, second ed., Graduate Texts in Mathematics, vol. 206, Readings in Mathematics, Springer, New York, 2008. MR 2376618 (2008j:11001)
- 10.
- M. Ram Murty and Kathleen L. Petersen, A Bombieri-Vinogradov theorem for all number fields, Trans. Amer. Math. Soc., to appear.
- 11.
- -, The generalized Artin conjecture and arithmetic orbifolds, Groups and symmetries, CRM Proc. Lecture Notes, vol. 47, Amer. Math. Soc., Providence, RI, 2009, pp. 259-265. MR 2500566
- 12.
- W. Narkiewicz, Units in residue classes, Arch. Math. (Basel) 51 (1988), no. 3, 238-241. MR 960401 (89k:11097)
- 13.
- Władysław Narkiewicz, Euclidean algorithm in small abelian fields, Funct. Approx. Comment. Math. 37 (2007), part 2, 337-340. MR 2363830 (2008j:11155)
- 14.
- Kathleen L. Petersen, Counting cusps of subgroups of
 , Proc. Amer. Math. Soc. 136 (2008), no. 7, 2387-2393. MR 2390505 (2008m:11091)
- 15.
- Pierre Samuel, About Euclidean rings, J. Algebra 19 (1971), 282-301. MR 0280470 (43:6190)
- 16.
- Peter J. Weinberger, On Euclidean rings of algebraic integers, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), Amer. Math. Soc., Providence, RI, 1973, pp. 321-332. MR 0337902 (49:2671)
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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
Posted:
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.
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