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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

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