The Factor Ring Structure of Quadratic Principal Ideal Domains

Abstract

AbstractPrevious authors have classified the possible factor rings of the Gaussian integers and the Eisenstein integers. Here, we extend this classification to the ring of integers of any quadratic number field, provided the ring has unique factorization. In the case of imaginary quadratic fields, the classification has exactly the same flavor as that of the Gaussian integers and Eisenstein integers. For real quadratic fields, the classification is only slightly more complicated.MSC: 11R1113F10 AcknowledgmentWe thank the anonymous referees for feedback which considerably improved the exposition of this article. We also thank the editors for their sharp eyes in proofing this article and for their help during the review process.Additional informationNotes on contributorsJohn GreeneJohn Greene received his Ph.D. from the University of Minnesota in 1983. He is a professor of mathematics at the University of Minnesota Duluth, where he has been for 30+ years. His research interests include special functions, combinatorics and (elementary) computational number theory.Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812jgreene@d.umn.eduWeizhi JingWeizhi Jing received his B.S. in mathematics as part of a joint degree program between the University of Minnesota Duluth and Shanxi University of China in 2019 and received his Master’s degree from UMD in 2021. He is dedicated to mathematical education and hopes to pursue a Ph.D. in ring theory or algebraic number theory.Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812jing0049@d.umn.edu