Good evening! I need your help. Let $p$ be a prime number and consider the ring $\mathbb{Z}_p = \mathbb Z/p\mathbb Z$.
(i.) Find all divisors of $0$ in $\mathbb{Z}_p$
(ii.) Show that for $0 \leq a$, where $a$ is less than p, the polynomial $f(x)=x^p-a \in \mathbb{Z}_p[x]$ has a linear factor in $\mathbb{Z}_p$.
For (i): Since $p$ is a prime then $\mathbb{Z}_p$ doesn’t contain a zero divisor. For (ii): If I choose $p=2$ and $a=1$, I see that $x^2-1$ has linear factors $x+1$ and $x-1$.May someone help me with a well procedure of proving (ii)?