高斯分圆公式[Gauss's Cyclotomic Formula] - 达人(其实～我是卧底，来自外星)

2005/06/09 | 高斯分圆公式[Gauss's Cyclotomic Formula]
类别(∑〖数学〗) | 评论(3) | 阅读(193) | 发表于 23:22: Let p > 3 be a prime number, then

where R(x,y) and S(x,y) are homogeneous polynomials in x and y with integer coefficients. Gauss (1965, p. 467) gives the coefficients of R and S up to p = 23.

Kraitchik (1924) generalized Gauss's formula to odd squarefree integers n > 3. Then Gauss's formula can be written in the slightly simpler form

where and have integer coefficients and are of degree and , respectively, with the totient function and a cyclotomic polynomial. In addition, is symmetric if n is even;otherwise it is antisymmetric. is symmetric in most cases, but it antisymmetric if n is of the form (Riesel 1994, p. 436). The following table gives the first few and s (Riesel 1994, pp. 436-442).
[left]n[/left]

[left]5[/left]
1

[left]7[/left]

[left]11[/left]; 0