Number fields without small generators

Martin Widmer, Jeffrey Vaaler

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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.
Original languageEnglish
Pages (from-to)379-385
Number of pages7
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number3
Early online date29 May 2015
Publication statusPublished - Nov 2015

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