Unit groups of quotient rings of complex quadratic rings

Abstract

For a square-free integer d other than 0 and 1, let \(K = \mathbb{Q}\left( {\sqrt d } \right)\), where \(\mathbb{Q}\) is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over \(\mathbb{Q}\). For several quadratic fields \(K = \mathbb{Q}\left( {\sqrt d } \right)\), the ring Rd of integers of K is not a unique-factorization domain. For d was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of Rd/ for the cases d = −2,−3.