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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: [email protected]
MARTIN WIDMER
Affiliation:
Department of Mathematics, Royal Holloway, University of London, TW20 0EX EghamUK e-mail: [email protected]
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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 

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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