Frontiers of Mathematics in China >
Unit groups of quotient rings of complex quadratic rings
Received date: 27 Nov 2015
Accepted date: 02 Mar 2016
Published date: 30 Aug 2016
Copyright
For a square-free integer d other than 0 and 1, let , where is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over . For several quadratic fields , the ring Rdof 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 was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of for the cases d = −2,−3.
Key words: Complex quadratic ring; quotient ring; unit group; quadratic field
Yangjiang WEI , Huadong SU , Gaohua TANG . Unit groups of quotient rings of complex quadratic rings[J]. Frontiers of Mathematics in China, 2016 , 11(4) : 1037 -1056 . DOI: 10.1007/s11464-016-0567-2
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