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}(\sqrt{d}), 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}(\sqrt{d}), the ring R_dof integers of K is not a unique-factorization domain. For d<0, there exist only a finite number of complex quadratic fields, whose ring Rd 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 Rd, and let n be an arbitrary positive integer. The unit groups of {{\displaystyle R}}_d/⟨v^n⟩ was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of {{\displaystyle R}}_d/⟨v^n⟩ for the cases d = −2,−3.

Key words： Complex quadratic ring; quotient ring; unit group; quadratic field

References

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