Finding Elements With Given Factorization Lengths and Multiplicities

Many algebraic number rings exhibit nonunique factorization of elements into irreducibles. Not only can the irreducibles in the factorizations be different, but the number of irreducibles in the factorizations can also vary. A basic question then is: Which sets can occur as the set of factorization lengths of an element? Moreover, how often can each factorization length occur? While these questions are most pertinent in algebraic number rings, their pertinence extends to Dedekind domains and a broader class of structures called Krull monoids. Surprisingly, for a large subclass of Krull monoids, Kainrath was able to resolve completely the question of which length sets and length multiplicities can be realized. In this article, we explain the context of Kainrath's theorem and give a constructive proof for an important case, namely Krull monoids with infinite nontorsion class group. We also construct length sets in a case not covered by Kainrath's theorem to illustrate the difficulty of the general problem.