Unit groups of quotient rings of complex quadratic rings

Research paper by Yangjiang Wei, Huadong Su, Gaohua Tang

Indexed on: 20 Jul '16Published on: 19 Jul '16Published in: Frontiers of Mathematics in China


For a square-free integer d other than 0 and 1, let \(K = \mathbb{Q}\left( {\sqrt d } \right)\), where Q is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over Q. For several quadratic fields \(K = \mathbb{Q}\left( {\sqrt d } \right)\), the ring R d of integers of K is not a unique-factorization domain. For d < 0, there exist only a finite number of complex quadratic fields, whose ring R d of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = −1,−2,−3,−7,−11,−19,−43,−67,−163. Let ϑ denote a prime element of R d , and let n be an arbitrary positive integer. The unit groups of R d /〈ϑ n 〉 was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of R d /〈ϑ n 〉 for the cases d = −2,−3.