摘要
For a square-free integer d other than 0 and 1, let K = Q（√d）, 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 = Q（√d）, the ring Rd 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 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 Q denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/（Q^n） was determined by Cross in 1983 for the case d = -1. This paper completely determined the unit groups of Rd/（Q^n） for the cases d = -2, -3.
For a square-free integer d other than 0 and 1, let K = Q（√d）, 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 = Q（√d）, the ring Rd 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 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 Q denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/（Q^n） was determined by Cross in 1983 for the case d = -1. This paper completely determined the unit groups of Rd/（Q^n） for the cases d = -2, -3.
基金
This work was supported by the National Natural Science Foundation of China （Grant Nos. 11461010, 11161006）, the Guangxi Natural Science Foundation （2014GXNSFAAll8005, 2015GXNSFAA139009）, the Guangxi Science Research and Technology Development Project （1599005-2-13）, and the Science Research Fund of Guangxi Education Department （KY2015ZD075）.
