r-循环矩阵和分圆类

r-Circulant matrices and the cyclotomy

研究了分圆类和r-循环矩阵(r>0)之间的关系,给出了分圆类、高斯周期和r-循环矩阵之间的一个对应;用一系列特殊的r-循环矩阵Hk解释了高斯周期,它们两者具有很多相似的性质,如乘积的线性性质等.此外还研究了Hk的周期多项式以及它们的逆对称问题,发现所有Hk具有相同的特征多项式,相同的特征值,且不赖于r和k的选取;找到了Hk满足逆对称性质的充分必要条件.这些都是非常有意义的结果.

This paper, makes a research on relations between the cyclotomic classes, Gaussian periods and rcirculant matrices, and create a correspondence between Gaussian periods and rcirculant matrices. It paraphrases Gaussian periods with a series of special rcirculant matrices Hk, and shows that they have some similar properties, such as the linear property of multiplication etc. Furthermore, It researches Gaussian periods polynomials of Hk and their inverse symmetry property and obtains that all Hk have the same characteristics and the same eigenvalues which are independent of the choice of k amd r; and find the sufficient and necessary condition of Hk's having the inverse symmetry property. All these results are all meaningful.