由欧拉公式,(x^n-1在复数域上的n个解为x_k=e^{i(frac{2kΠ}{n})}),k=0,1,2,.....,n-1
其中(x_k=cosfrac{2kΠ}{n}+isinfrac{2kΠ}{n},k=0,1,2,.....,n-1)
(x^n-1)在复数域上的标准分解为(x^n-1=(x-x_0)(x-x_1).....(x-x_{n-1}))
经观察,((x-x_q)(x-x_{n-q})=x^2+1-2xcosfrac{2qΠ}{n}),为实数域上的既约多项式
当n为奇数时,(x_0=1,则x^n-1=(x-1)[x^2+1-2xcosfrac{2Π}{n}][x^2+1-2xcosfrac{4Π}{n}].....[x^2+1-2xcosfrac{(n-1)Π}{n}])
当n为偶数时,(x_0=1,x_{n/2}=-1,则x^n-1=(x-1)(x+1)[x^2+1-2xcosfrac{2Π}{n}][x^2+1-2xcosfrac{4Π}{n}].....[x^2+1-2xcosfrac{(n-2)Π}{n}])