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 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.