Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 2, pp. 425-445.

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant

Δ=7 2 ,9 2 ,13 2 ,19 2 ,31 2 ,37 2 ,43 2 ,61 2 ,67 2 ,103 2 ,109 2 ,127 2 ,157 2 .

A large part of the proof is in establishing the following more general result: Let K be a Galois number field of odd prime degree and conductor f. Assume the GRH for ζ K (s). If

38(-1) 2 (logf) 6 loglogf<f,

then K is not norm-Euclidean.

En supposant que l’hypothèse de Riemann généralisée (HRG) soit vérifiée, nous montrons que les corps de nombres galoisiens de degré 3 qui sont euclidiens pour la norme sont précisément ceux dont le discriminant est l’un des entiers suivants :

Δ=7 2 ,9 2 ,13 2 ,19 2 ,31 2 ,37 2 ,43 2 ,61 2 ,67 2 ,103 2 ,109 2 ,127 2 ,157 2 .

Une grande partie de la preuve consiste à établir le résultat plus général suivant : soit K un corps de nombres galoisien de degré premier impair et de conducteur f. Supposons que HRG soit vérifiée pour ζ K (s). Si

38(-1) 2 (logf) 6 loglogf<f,

alors K n’est pas euclidien pour la norme.

DOI: 10.5802/jtnb.804
Classification: 11A05, 11R04, 11R16, 11R80, 11Y40
Keywords: norm-Euclidean, Galois fields, cubic fields, GRH, Dirichlet characters
McGown, Kevin J. 1

1 Department of Mathematics University of California, San Diego La Jolla, California, 92093, USA Current address : Department of Mathematics Oregon State University 368 Kidder Hall Corvallis, Oregon, 97331, USA
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McGown, Kevin J. Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 2, pp. 425-445. doi : 10.5802/jtnb.804. http://archive.numdam.org/articles/10.5802/jtnb.804/

[1] N. C. Ankeny, The least quadratic non residue. Ann. of Math. (2) 55 (1952), 65–72. | MR | Zbl

[2] E. Bach, Explicit bounds for primality testing and related problems. Math. Comp. 55 (1990), no. 191, 355–380. | MR | Zbl

[3] H. Davenport, Multiplicative number theory, third edition. Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000, Revised and with a preface by Hugh L. Montgomery. | MR | Zbl

[4] H. J. Godwin and J. R. Smith, On the Euclidean nature of four cyclic cubic fields. Math. Comp. 60 (1993), no. 201, 421–423. | MR | Zbl

[5] H. Heilbronn, On Euclid’s algorithm in cubic self-conjugate fields. Proc. Cambridge Philos. Soc. 46 (1950), 377–382. | MR | Zbl

[6] Y. Ihara, V. K. Murty, and M. Shimura, On the logarithmic derivatives of Dirichlet L-functions at s=1. Acta Arith. 137 (2009), no. 3, 253–276. | MR

[7] J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem. Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, pp. 409–464. | MR | Zbl

[8] E. Landau, Zur Theorie der Heckeschen Zetafunktionen, welche komplexen Charakteren entsprechen. Math. Z. 4 (1919), no. 1-2, 152–162. | MR

[9] F. Lemmermeyer, The Euclidean algorithm in algebraic number fields. Exposition. Math. 13 (1995), no. 5, 385–416. | MR | Zbl

[10] K. J. McGown, Norm-Euclidean cyclic fields of prime degree. Int. J. Number Theory (to appear). | MR

[11] H. L. Montgomery, Topics in multiplicative number theory. Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin, 1971. | MR | Zbl

[12] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, second edition. Springer-Verlag, Berlin, 1990. | MR | Zbl

[13] A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Sém. Théor. Nombres Bordeaux (2) 2 (1990), no. 1, 119–141. | Numdam | MR | Zbl

[14] G. Poitou, Minorations de discriminants (d’après A. M. Odlyzko). Séminaire Bourbaki, Vol. 1975/76 28ème année, Exp. No. 479, Springer, Berlin, 1977, pp. 136–153. Lecture Notes in Math., Vol. 567. | Numdam | MR | Zbl

[15] G. Poitou, Sur les petits discriminants. Séminaire Delange-Pisot-Poitou, 18e année: (1976/77), Théorie des nombres, Fasc. 1, Secrétariat Math., Paris, 1977, pp. Exp. No. 6, 18. | Numdam | MR | Zbl

[16] H. M. Stark, Some effective cases of the Brauer-Siegel theorem. Invent. Math. 23 (1974), 135–152. | MR | Zbl

[17] H. M. Stark, The analytic theory of algebraic numbers. Bull. Amer. Math. Soc. 81 (1975), no. 6, 961–972. | MR | Zbl

[18] A. Weil, Sur les “formules explicites” de la théorie des nombres premiers. Comm. Sém. Math. Univ. Lund (1952), 252–265. | MR | Zbl

[19] A. Weil, Sur les formules explicites de la théorie des nombres. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 3–18. | MR | Zbl

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