Let $L\subset \Bbb{C}$ be a lattice, assume there is a $\gamma \in \Bbb{C}$ such that $\gamma L = L$, prove the following two facts:
(1) $\gamma$ must be roots of unity
(2) if $\gamma$ is not real, then for any element $\ell \in L$ with minimal length, we can generate the lattice using $\ell, \gamma\ell$.
My attempt:
first I prove $\|\gamma\| = 1$ :assume $m\in L$ has minimal length, then $\|\gamma m\| \ge \|m\|$ which implies $\|\gamma\|\ge 1$ similarily assume $\gamma n$ has minimal length then $\|n\|\ge \|n\gamma\|$ therefore $\|\gamma\| = 1$.
Second , if it's not roots of unity, then assume minimal length is $k$ then the points on the lattice will dense on the circle with radius $k$ (by repeating apply $\gamma$) that's not possible by change of coordinate assume generator are $(a,0)$ and $(p,q)$ ($a\ne 0, q\ne 0$) then there are infinite many point of the form $(na+mp,mq)$ with $m,n\in \Bbb{Z}$ lies on the circle, but the norm of them $(na+mp)^2 +(mq)^2\to \infty$ (since there are infinite many points $m \to \infty$ or $n\to \infty$ , if $m\to \infty$ the second term will goes to infty, if not the first term will goes to infty)
And I don't know how to prove (2)?