Abstract: Let f=Ord_p(q), ε_pbe a primitive p-th root of unity，γ∈Z[ε_p]。Suppose that f is odd and γγ^-=n（where n∈Z, q|n）. By some assumption, K.T.Arasu and A.Pott obtain that γ can be written as the form γ＝∑^e_i=1a∑^f-1_j=0ε^iqj_p(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： 设f=Ord_p(q), ε_p为本原p次单位方根，γ∈Z[ε_p]。若f是奇数且γγ^-=n（其中n∈Z, q|n）,加上别的一些条件，Arasu和Pott证明了γ具有形式：γ＝∑^e_i=1a∑^f-1_j=0ε^iqj_p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