回复 2# hbghlyj
(3)$L_{a,b}$是a,b的整系数线性组合,它对加减法封闭,$\emptyset\ne L\subset \mbb C,0\ne z\in \mbb C$,令zL={zu|$u\in L$},$R_{a,b}$={$z|z\in\mbb C^*\wedge zL=L$}
,可证:若$z_1,z_2\in R_{a,b}$,则$z_1z_2\in R_{a,b}$
$z_1z_2L_{a,b}=z_1(z_2L_{a,b})=z_1L_{a,b}=L_{a,b}$
可证:若$z\in R_{a,b}$,则|z|=1
设m=$\min\{|B||B\in L_{a,b}\wedge B\ne0\}$
则|z|m=m,|z|=1
可证:$|R_{a,b}|\leq6$
设m=|A|,任取$z_1,z_2\in R_{a,b}$,则$z_1A,z_2A\in R_{a,b}$,则$|z_1-z_2|\geq m$,$\arg(\frac{z_1}{z_2})\geq\frac{\pi}{3}$
$R_{a,b}$是有限集,由(2)可证:$|R_{a,b}|=n$
因为$1,-1\in R_{a,b}$,所以n=2,4,6
$R_{a,b}$的所有可能为
{1,-1}{1,-1,i,-i}$\{1,\omega ,\omega ^2\cdots\omega^5\}$ |