The number of units in the quotient ring $\Bbb Z_5[x]/(x^4-1)$

Question

I am asked to find the number of units in the quotient ring $\Bbb Z_5[x]/(x^4-1)$, where $\Bbb Z_5$ is the finite field consisting of 5 elements. I know that this ring has $5^4$ elements (which is not quite small), so I think I can't just do this using only computation. Any hints?

Answer 1

Hint: Factor $x^4-1$ in $\Bbb Z_5[x]$ and use the Chinese remainder theorem.