一类四边形的有理距点问题

On rational distance point problem for quadrilaterals

通过指出杜和邓的证明过程的失误,说明Steinhaus整距点问题至今仍然是没有解决的公开问题.利用数域的扩张和Galois群的一些结论,研究了边长的平方为有理数,且一组对角之和为2kπ/n的四边形,其中k<n,gcd(k,n)=1,n≥7,n≠8,10,12.证明了平面上不存在这样的点,到该四边形的4个顶点的距离的平方均为有理数.

In the present paper, by pointing out some errors of Du and Deng, we know that the wellknown integer distance point problem of Steinhaus is still open. Using the result of Galois extension,we discuss the Steinhaus problem for some quadrilaterals. In fact, we show that for the case of quadrilaterals which the sum of a pair of corner is 2kπ/n where n≥7, k 〈 n, gcd（k, n） = 1, and n ≠ 8, 10, 12,there does not exist points in the plane such that the squares of distances between this point to all the four vertices of the quadrilaterals are rational number simultaneously.