Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T03:22:22.413Z Has data issue: false hasContentIssue false
  • English
  • Français

Number fields without small generators

Published online by Cambridge University Press:  29 May 2015

JEFFREY D. VAALER  and
MARTIN WIDMER
JEFFREY D. VAALER
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712 e-mail: vaaler@math.utexas.edu
MARTIN WIDMER
Affiliation:
Department of Mathematics, Royal Holloway, University of London, TW20 0EX EghamUK e-mail: martin.widmer@rhul.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Let D > 1 be an integer, and let b = b(D) > 1 be its smallest divisor. We show that there are infinitely many number fields of degree D whose primitive elements all have relatively large height in terms of b, D and the discriminant of the number field. This provides a negative answer to a question of W. Ruppert from 1998 in the case when D is composite. Conditional on a very weak form of a folk conjecture about the distribution of number fields, we negatively answer Ruppert's question for all D > 3.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 3 , November 2015 , pp. 379 - 385
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bhargava, M. The density of discriminants of quintic rings and fields. Ann. of Math. 172 (2010), 1559–1591.Google Scholar
[2] Bombieri, E. and Gubler, W. Heights in Diophantine Geometry. (Cambridge University Press, 2006).Google Scholar
[3] Datskovsky, B. and Wright, D. J. Density of discriminants of cubic field extensions. J. Reine Angew. Math. 386 (1988), 116–138.Google Scholar
[4] Cohen, H. Diaz, F. Diaz, Y. and Olivier, M. Enumerating quartic dihedral extensions of $\mathbb{Q}$ . Comp. Math. 133 (2002), 65–93.Google Scholar
[5] Ellenberg, J. and Venkatesh, A. The number of extensions of a number field with fixed degree and bounded discriminant. Ann. of Math. 163 (2006), 723–741.Google Scholar
[6] Northcott, D. G. An inequality in the theory of arithmetic on algebraic varieties. Proc. Camb. Phil. Soc. 45 (1949), 502–509 and 510–518.Google Scholar
[7] Roy, D. and Thunder, J. L. A note on Siegel's lemma over number fields. Monatsh. Math. 120 (1995), 307–318.Google Scholar
[8] Ruppert, W. Small generators of number fields. Manuscripta Math. 96 (1998), 17–22.Google Scholar
[9] Schmidt, W. M. Northcott's Theorem on heights I. A general estimate. Monatsh. Math. 115 (1993), 169–183.Google Scholar
[10] Silverman, J. Lower bounds for height functions. Duke Math. J. 51 (1984), 395–403.Google Scholar
[11] Vaaler, J. D. and Widmer, M. A note on small generators of number fields. Diophantine Methods, Lattices and Arithmetic Theory of Quadratic Forms. Contemp. Math. vol. 587 (Amer. Math. Soc., Providence, RI, 2013).Google Scholar
[12] Widmer, M. Counting points of fixed degree and bounded height. Acta Arith. 140.2 (2009), 145–168.Google Scholar