The problem statement, all variables and given/known data Prove that the ideal I = [tex]\langle x^2 + 1 \rangle[/tex] is prime in Z[x] but not maximal. The attempt at a solution I'm having a hard time doing this because Z[x] is not a field. I know that x2 + 1 is irreducible in Z[x] so the proof must hinge on this fact. Let f(x) and g(x) belong to Z[x] and suppose f(x)g(x) is in I. Then there is some q(x) in Z[x] such that f(x)g(x) = (x2 + 1)q(x). How can I show that either f(x) or g(x) belongs to I? How does the irreducibility of x2 + 1 come into play here?