Is $x^4 + 4$ irreducible in $\mathbb{Z}_5$? Well, I'm having doubts, isnt that $\mathbb{Z}_5$ has no zero divisors, and now you cant factor  $x^4 + 4$ ?

\begin{align}
x^4+4&=x^4-1\\
&=(x^2-1)(x^2+1)\\
&=(x-1)(x+1)(x^2-4)\\
&=(x-1)(x+1)(x+2)(x-2)\\
&=(x-1)(x-2)(x-3)(x-4).
\end{align}
