all prime elements of $ \Bbb{Z}_p$ 
What are all prime elements of $\Bbb{Z}_p$ ?

$\Bbb{Z}_p$ is a local ring with unique maximal ideal $(p)$.
Prime element is generator of $(p)$, so $p$ and $-p$ are prime elements.
What about other prime elements ?
I couldn't come up with other prime elements of $\Bbb{Z}_p$. All prime elements are $±p$?
Thank you in advance.

This is equivalent to finding all units of $\mathbb Z_p$, and since $\mathbb Z_p$ is local, $$\mathbb Z_p^{\times} = \cup_{x\in\{1, 2, \dots, p-1\}} x + p\mathbb Z_p.$$
In particular, $1+p+p^2+p^3+\cdots=\frac{1}{1-p}$ is a unit, and $(\frac{p}{1-p})=(p)$.
