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 . 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 .

As for , what are the prime and maximal ideals, and more importantly, how in the world do we know that we have found them all?