Growth results and Euclidean ideals

Lenstra generalized the Euclidean algorithm via his concept of Euclidean ideals. While a domain with a Euclidean algorithm has trivial class group, a domain with a Euclidean ideal has cyclic class group. This paper generalizes Harperʼs variation of Motzkinʼs Lemma to the Euclidean ideal situation and then uses the Large Sieve to obtain growth results. It concludes that if a certain set of primes in a number field with cyclic class group is large enough, then said field has a Euclidean ideal.