I want to answer part(b) here in this question:
Let $C^{*}$ denote the group of nonzero complex numbers under multiplication, and $S^{1} \subset C^{*}$ the subgroup of complex numbers of length one. Torsion elements of $C^{*}$ are called roots of unity.
(a) Show that $Tor(C^{*}) \subset S^1.$ Now give a simple reason that $Tor(C^{*}) \neq S^1.$
(b) Define $z \in S^{1} $ by $z = \frac{3}{5} + i \frac{4}{5}.$ Show that $z \notin Tor(\mathbb{C^*}).$
My question is:
I get this hint to answer it: "Show that the real and imaginary parts of $(3 + 4i)^n$ are congruent to 3 and 4 modulo 5, respectively."
But I do not understand how proving this hint will answer the question? could anyone help me in answering this question, please?
Note that: part $(a)$ answer can be found here A simple reason that $Tor(C^{*}) \neq S^1.$
Any $z\in \Bbb S^1$ has modulus 1 and thus can be written as $$ z=e^{i\theta}. $$
Such a number is a root of unity if $\exists n$ such that $z^n=e^{i\theta n}=1$. This occurs if and only if $n\theta\in2\pi\Bbb Z$.
Can you conclude?
It is easy to find a real number $\theta$ such that $n\theta$ doesn't belong to $2\pi\Bbb Z$. That is, the equality $$ n\theta=2\pi m $$ is never true, whatever $n,m\in\Bbb Z$ you take.
For example $\theta=1$ does the job.
So for such $\theta$, one never gets $$ z^n=1 $$ so, such a $z=e^{i\theta}$ doesn't belong to $\operatorname{Tor}(\Bbb C^*)$, but stay obviously in $\Bbb S^1$.