Acta Scientiarum Naturalium Universitatis Pekinensis

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Impossibility of Difference Sets Depending on Certain Cyclotomic Equation

WAN Zhaoze   

  1. School of Mathematical Sciences, Peking University, 100871
  • Received:1997-09-01 Online:1999-01-20 Published:1999-01-20



  1. 北京大学数学科学学院,北京,100871

Abstract: Let f=Ordp(q), εpbe a primitive p-th root of unity,γZ[εp]。Suppose that f is odd and γγ-=n(where nZ, q|n). By some assumption, K.T.Arasu and A.Pott obtain that γ can be written as the form γ=∑ei=1af-1j=0εiqjp(where p-1=ef). And note that it would be nice if the condition "f is odd" could be removed. In this paper, the author will deal with the case "f is even" and get a corresponding results.

Key words: cyclotomic fields, difference sets, prime ideal, Galois group

摘要: f=Ordp(q), εp为本原p次单位方根,γZ[εp]。若f是奇数且γγ-=n(其中nZ, q|n),加上别的一些条件,Arasu和Pott证明了γ具有形式:γ=∑ei=1af-1j=0εiqjp其中p-1=ef, 并且指出如果条件“f是奇数”能去掉的话,将是一个很好的结果,本文作者研究了f为偶数的情形并得到相应的结论。

关键词: 分圆域, 差集, 素理想, Galois群

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