Number Fields without Small Generators

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 questions 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.