Let p be a prime integer. Prove that [1] and [p-1] are the only elements in Zp that are their own multiplicative inverses.
Hint: If $\displaystyle x^2 \equiv 1 mod (p)$, then $\displaystyle x \equiv +1$ or $\displaystyle x \equiv -1$.
$\displaystyle U_p=\{[1], [2], \cdots ,[p-1]\}$ for a prime number p > 2 and $\displaystyle U_2=\{1\}$, where $\displaystyle U_p$ is the group of units in $\displaystyle \mathbb{Z}_p$.