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**Homework Statement**

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 x

^{2}+ 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) = (x

^{2}+ 1)q(x). How can I show that either f(x) or g(x) belongs to I? How does the irreducibility of x

^{2}+ 1 come into play here?