Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory

Abstract

Let D be the ring of integers in a finite extension of the rationals. The classic examination of the factorization properties of algebraic integers usually begins with the study of norms. In this paper, we show using the ideal class group, C(D), of D that a deeper examination of such properties is possible. Using the class group, we construct an object known as a block monoid, which allows us to offer proofs of three major results from the theory of nonunique factorizations: Geroldinger's theorem, Carlitz's theorem, and Valenza's theorem. The combinatorial properties of block monoids offer a glimpse into two widely studied constants from additive number theory, the Davenport constant and the cross number. Moreover, block monoids allow us to extend these results to the more general classes of Dedekind domains and Krull domains.

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Notes on contributors

Paul Baginski

SCOTT T. CHAPMAN received his B.A. from Wake Forest University in 1981, his M.S. from the University of North Carolina at Chapel Hill in 1984 and his Ph.D. from the University of North Texas in 1987. He is currently the Scholar in Residence at Sam Houston State University in Huntsville, Texas. He is serving as Editor-Elect of the American Mathematical Monthly during 2011 and will serve a five-year term as Editor starting in 2012.

Scott T. Chapman

PAUL BAGINSKI received his B.S. and M.S. from Carnegie Mellon University in 2003 and his Ph.D. from the University of California, Berkeley in 2009. He is currently an NSF International Research Fellow, fulfilling his postdoctoral appointment at the Institut Camille Jordan at Université Lyon 1 in France. His nonmathematical hobbies include travel, cooking, and cinema, especially from the silent era.