Existence of square root in $\mathbb Z_n$?

Question

I had this question on my final exam and I struggled with it.

It asks to prove or disprove the following:

$$\forall m \in Z, \ \forall \ [a] \in Z_{m}, \ \exists \ [b] \in Z_{m}, [a]=[b]^{2} $$

I claimed that it's true, and wrote that for an arbitrary $m$, and an arbitrary $a$, $[a] = [b]^{2}$ is equivalent to solving $x \equiv b^{2} \pmod{m}$ which is possible since $\gcd(m,1) \mid b^2$ .

Answer 1

No way. In $\mathbb{Z}_3$, the equation $x^2 = \hat 2$ has no solution.

Answer 2

In words, this is asking if every element of $\mathbb{Z}_m$ is a perfect square. This is false, and can be shown by a counterexample.

Take $m=3$, and consider the congruence class $a=[2]$. Since $[0]^2 = [0], [1]^2=[1]$, and $[2]^2 = [1]$, there is no $b\in\mathbb Z_m$ such that $a = b^2$.