I'm working on an exam that Michael Artin once gave, where one of the questions is basically, Consider the homomorphism from Z[x] to Z given by x --> i. What does this homomorphism tell you about the ideals of Z? So far I haven't come up with anything. I know in advance that the ideals of the Gaussian integers are principal, but I don't see how I can prove that just by looking at the given homomorphism. I know that there is a bijective correspondence between ideals of Z[x] containing x^2 + 1 and ideals of Z, but I'm stuck. For example, if I look at the ideal generated by x^2 + 1 and x+1, how do I know whether this is a proper ideal or not? If it is proper, then I know that the ideal generated by i + 1 is a proper ideal of Z. But I don't know how to tell whether or not the ideal generated by x^2 + 1 and x+1 is proper. Do you guys think I'm going about this problem the right way? I've spent like half an hour thinking about it and am not making good progess.