h-ful ideals of bounded norm in number fields

Abstract

Consider a number field with its ring of integers. Call an ideal of the ring of integers h-ful if all exponents in its prime factorization are at least h. In this thesis, we will find an asymptotic expression with one main term for the number of h-ful ideals of norm at most x. To do this, we will use two papers, one by Erdös and Szekeres, and one by Bateman and Grosswald, which have explored this problem in the rational numbers. We will adapt their methods to arbitrary number fields.

Furthermore, we will consider elements of the ring of integers where the principal ideal generated by them is h-ful. We will explore but not fully solve the problem of finding the number of such elements of height at most x, with a height function that is the product over all infinite places of the maximum of 1 and the valuation associated with that place.