all prime elements of $ \Bbb{Z}_p$

Question

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.

Answer 1

If $p$ is a prime element in a domain $A$, then all elements of the form $up$, with $u$ a unit in $A$, are prime as well and $(p)=(up)$.

If $A$ is local and its unique maximal ideal is principal generated by $p$, then the elements listed above are the only prime elements.

Indeed if $q$ is a prime element, we have $(q)\subseteq(p)$ and so $q=ap$ for some $a\in A$. Since $q$ is prime, we have either $a\in(q)$ or $p\in(q)$. In the latter case, $p=bq$ and we obtain $q=abq$, so $ab=1$ and $b$ is a unit. In the former case $a=cq$, so $1=cp$, which is a contradiction.

(By definition, prime elements are nonzero and noninvertible.)

Answer 2

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 $(\frac{p}{1-p})=(p)$.