Growth Results and Euclidean Ideals

TL;DR

This paper extends the theory of Euclidean ideals in number rings, demonstrating that large enough prime sets imply the existence of Euclidean ideals in rings with cyclic class groups, using advanced sieve methods.

Contribution

It generalizes Harper's Motzkin's lemma to Euclidean ideals and applies the large sieve to establish growth results for Euclidean ideals in number fields.

Findings

Large prime sets imply Euclidean ideals in cyclic class group rings

Generalization of Motzkin's lemma to Euclidean ideals

Application of large sieve to number field rings

Abstract

Lenstra's concept of Euclidean ideals generalizes the Euclidean algorithm; a domain with a Euclidean ideal has cyclic class group, while a domain with a Euclidean algorithm has trivial class group. This paper generalizes Harper's variation of Motzkin's lemma to Lenstra's concept of Euclidean ideals and then uses the large sieve to obtain growth results. It concludes that if a certain set of primes is large enough, then the ring of integers of a number field with cyclic class group has a Euclidean ideal.