RESEARCH ARTICLE

Unit groups of quotient rings of complex quadratic rings

  • Yangjiang WEI ,
  • Huadong SU ,
  • Gaohua TANG
Expand
  • School of Mathematics and Statistics Sciences, Guangxi Teachers Education University, Nanning 530023, China

Received date: 27 Nov 2015

Accepted date: 02 Mar 2016

Published date: 30 Aug 2016

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

For a square-free integer d other than 0 and 1, let K=(d), where is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over . For several quadratic fields K=(d), 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 Rd/vn was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of Rd/vn for the cases d = −2,−3.

Cite this article

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

1
Bini B, Flamini F. Finite Commutative Rings and Their Applications. Dordrecht: Kluwer Academic Publishers, 2002

DOI

2
Cross J T. The Euler φ-function in the Gaussian integers. Amer Math Monthly, 1983, 90: 518–528

DOI

3
Karpilovsky G. Units Groups of Classical Rings. New York: Oxford University Press, 1988

4
Pezda T. Cycles of polynomial mappings in two variables over rings of integers in quadratic fields. Cent Eur J Math, 2004, 2(2): 294–331

DOI

5
Pezda T. Cycles of polynomial mappings in several variables over rings of integers in finite extensions of the rationals II. Monatsh Math, 2005, 145: 321–331

DOI

6
Stark H M. A complete determination of the complex quadratic fields of class-number one. Michigan Math J, 1967, 14(1): 1–27

DOI

Outlines

/