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ON THE EXCEPTIONAL fieldS FOR A CLASS OF REAL quadratic fieldS

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Acta Mathematica Scientia 2011年第3期31卷 1179-1188页

作者：刘丽陆洪文 School of Mathematics Hefei University of Technology Department of Mathematics Tongji University

In this paper, we give a lower bound exp（2.2 × 10~8 ） for those discriminants of real quadratic fields Q（√ d） with d= N^2-4 and h（d）=1.

关键词： quadratic field class number discriminant Zeta-function lower bound

Square-Free Integers as Ideal Norms

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Acta Mathematica Sinica,English Series 2010年第4期26卷 621-628页

作者： Shigeru KANEMITSU Antanas LAURINCIKAS Jing MA Graduate School of Advanced Technology Kinki University Iizuka Fukuoka 820-8555 Japan Department of Mathematics and Informatics Vilnius University Naugarduko 24 LT 03225 Vilnius Lithuania Department of Mathematics Jilin University Changchun 130012 P. R. China School of Mathematics and System Sciences Shandong University Jinan 250100 P. R. China

Elaborating on Katzel＇s result, Zhai obtained a sharp short interval result for the number of ways of expressing a square-free integer as sums of two squares. In this note we view Zhai＇s decom- position as one by de... 详细信息

关键词： square-free integer ideal norm quadratic field

Unit groups of quotient rings of complex quadratic rings

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Frontiers of Mathematics in China 2016年第4期11卷 1037-1056页

作者： Yangjiang WEI Huadong SU Gaohua TANG School of Mathematics and Statistics Sciences Guangxi Teachers Education University Nanning 530023 China

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.

关键词： Complex quadratic ring quotient ring unit group quadratic field

The Quotient Ring of Algebraic Interger Ring

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Chinese Quarterly Journal of Mathematics 1995年第2期 77-80页

作者：丁瑞香刘广亮

ＴｈｅＱｕｏｔｉｅｎｔＲｉｎｇｏｆＡｌｇｅｂｒａｉｃＩｎｔｅｒｇｅｒＲｉｎｇ￥ＤｉｎｇＲｕｉｘｉａｎｇ；ＬｉｕＧｕａｎｇｌｉａｎｇ（ＰｕｙａｎｇＥｄｕｃａｔｉｏｎｓＣｏｌｌｅｇｅ，Ｈｅｎａｎ，４５７０００）Ａｂｓｔｒ... 详细信息

ＴｈｅＱｕｏｔｉｅｎｔＲｉｎｇｏｆＡｌｇｅｂｒａｉｃＩｎｔｅｒｇｅｒＲｉｎｇ￥ＤｉｎｇＲｕｉｘｉａｎｇ；ＬｉｕＧｕａｎｇｌｉａｎｇ（ＰｕｙａｎｇＥｄｕｃａｔｉｏｎｓＣｏｌｌｅｇｅ，Ｈｅｎａｎ，４５７０００）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｅｐａｐｅｒ，ｗｅ...

关键词： quadratic field algebraic intelger algebraic interger ring quotient ring

quadratic Number fields with Class Numbers Divisible by a Prime q

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Tsinghua Science and Technology 2004年第4期9卷 475-481页

作者：杨东张贤科 Department of Mathematical Sciences Tsinghua University Beijing 100084 China

Let q 5 be a prime number. Let k d=() be a quadratic number field, where d =--gqq(1)2(1) ---qqqquwuq12((1)+). Then the class number of k is divisible by q for certain integers u,w. Conversely, assume W / k is an unramified cyclic extension of degree q (which implies the class number of k is divisible by q), and W is the splitting field of some irreducible trinomial f(X) = XqaXb with integer coefficients, k Df=(())with D(f) the discriminant of f(X). Then f(X) must be of the form f(X) = Xquq2wXuq1 in a cer-tain sense where u,w are certain integers. Therefore, k d=() with d =-----qqqqqquwuq(1)122(1)((1)+). Moreover, the above two results are both generalized for certain kinds of general polynomials.

关键词： quadratic field class number unramified Newton抯 polygon