Linear Algebra Fields question I have the following statement (which is false) and I'm trying to understand why, I can't find a concrete example.
If $a$ belongs to $Z_n$ and $a^2 = 1$ then $a=1$ or $a=-1$
Can someone give me a direction?
Guy

The statement does hold when the structure is a field, i.e. when $n$ is prime. The statement is false for some composite $n$. For example, consider any $n$ of the form $x(x+2)$ for some $x\in\mathbb{N}^+$ then we will have
$$(x+1)^2 \equiv x^2 + 2x + 1 \equiv n + 1 \equiv 1 \pmod n$$
so that $\pm(x+1)$ is a solution in addition to $\pm1$.
