I'm working on prime an maximal ideals. My partner and I are studying for our final exam and got conflicting answers.

The question was to find all of the prime and maximal ideals of $\displaystyle \mathbb Z_7$. My answer was that because a finite integral domain is a field, the prime and maximal ideals coincide, but that there are no prime and maximal ideals for $\displaystyle \mathbb Z_7$.

As for $\displaystyle \mathbb Z_3 \times \mathbb Z_5$ , what are the prime and maximal ideals, and more importantly, how in the world do we know that we have found them all?