Finding Elements With Given Factorization Lengths and Multiplicities

Abstract

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.

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Notes on contributors

Paul Baginski

PAUL BAGINSKI received his Ph.D. from the University of California, Berkeley in 2009. He spent an NSF postdoc at Université Lyon I in France, followed by a visiting position at Smith College. He has been an assistant professor of mathematics at Fairfield University since 2013.

Ryan Rodriguez

RYAN RODRIGUEZ received his mathematics Ph.D. in 2014 from the University of California, San Diego.

George J. Schaeffer

GEORGE J. SCHAEFFER received his B.S. and M.S. in mathematical sciences at Carnegie Mellon University in 2007 and his Ph.D. from the University of California, Berkeley in 2012. He is currently a lecturer in the Department of Mathematics at Stanford University.

Yiwei She

YIWEI SHE is a Prize Postdoctoral Fellow at Columbia University. She received her B.S. from Northwestern University in 2010 and her Ph.D. from the University of Chicago in 2015.