Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-18T20:36:48.819Z Has data issue: false hasContentIssue false
  • English
  • Français

ℤ[] is Euclidean

Published online by Cambridge University Press:  20 November 2018

Malcolm Harper
Malcolm Harper*
Affiliation:
Champlain College, St. Lambert, Québec, J4P 3P2 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide the first unconditional proof that the ring $\mathbb{Z}[\sqrt{14}]$ is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of $\mathbb{Q}$. It is proved that if $K$ is a real quadratic field (modulo the existence of two special primes of $K$) or if $K$ is a cyclotomic extension of $\mathbb{Q}$ then:

the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain.

The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when $K$ is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property.

Keywords

11R0411R11
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 1 , 01 February 2004 , pp. 55 - 70
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Bombieri, E., Friedlander, J. B. and Iwaniec, H., Primes in arithmetic progressions to large modulii. Acta Math. 156(1986), 203–251, MR 88b:11058.Google Scholar
[2] Clark, David A., The Euclidean algorithm for Galois extensions of the rational numbers. Ph.D. thesis, McGill University, Montreal, 1992.Google Scholar
[3] Clark, David A. and Murty, M. Ram, The Euclidean algorithm for Galois extensions of ℚ . J. Reine Angew. Math. 459(1995), 151–162, MR 96h:11104.Google Scholar
[4] Fouvry, Étienne, Théorème de Brun-Titchmarsh; application au théorème de Fermat. Invent. Math. 79(1985), 383–407, MR 86g:11052.Google Scholar
[5] Gupta, Rajiv and Murty, M. Ram, A remark on Artin's conjecture. Invent.Math. 78(1984), 127–130, MR 86d:11003.Google Scholar
[6] Gupta, Rajiv, Murty, M. Ram and Murty, V. Kumar, The Euclidean algorithm for S-integers. In: Number Theory (Montreal, June 1985), CMS Conf. Proc. 7, Amer. Math. Soc., 1987, 189–201, MR 88h:11088.Google Scholar
[7] Harper, Malcolm, A family of Euclidean rings containing Z[p14]. CMS talk, December 1998.Google Scholar
[8] Harper, Malcolm, A proof that ℤ[] is Euclidean. Ph.D. thesis, McGill University, Montreal, 2000.Google Scholar
[9] Harper, Malcolm and Murty, M. Ram, Euclidean rings of algebraic integers. Canad. J. Math. 56(2004), 71–76.Google Scholar
[10] Heath-Brown, D. R., Artin's conjecture for primitive roots. Quart. J. Math. Oxford Ser. (2) 37(1986), 27–38, MR 88a:11004.Google Scholar
[11] Hooley, Christopher, On Artin's conjecture. J. Reine Angew.Math. 225(1967), 209–220, MR 34 #7445.Google Scholar
[12] Iwaniec, Henryk, A new form of the error term in the linear sieve. Acta Arith. 37(1980), 307–320, MR 82d:10069.Google Scholar
[13] Lenstra, Hendrik W. Jr., Euclid's algorithm in cyclotomic fields. J. London Math. Soc. (2) 10(1975), 457–465, MR 52 #8100.Google Scholar
[14] Lenstra, Hendrik W. Jr., Quelques exemples d'anneaux euclidiens. C. R. Acad. Sci. Paris Sér. D 286(1978), 683–685.Google Scholar
[15] Lenstra, Hendrik W. Jr., Euclidean number fields I. Math. Intelligencer 2(1979), 6–15, MR 81b:12002.Google Scholar
[16] Motzkin, Th., The Euclidean algorithm. Bull. Amer. Math. Soc. 55(1949), 1142–1146.Google Scholar
[17] Ojala, T., Euclid's algorithm in the cyclotomic field ℚ(ζ16) . Math. Comp. 31(1977), 268–273, MR 54 #10194.Google Scholar
[18] Samuel, Pierre, About Euclidean rings. J. Algebra 19(1971), 282–301, MR 43 #6190.Google Scholar
[19] Weinberger, Peter J., On Euclidean rings of algebraic integers. In: Analytic Number Theory (St. Louis, 1972), Proc. Sympos. Pure Math. XXIV, Amer. Math. Soc., 1973, 321–332, MR 49 #2671.Google Scholar
[20] Wilson, Robin J., The large sieve in algebraic number fields. Mathematika 16(1969), 189–204, MR 41 #8374.Google Scholar