Re: list all maximal ideals

The key to all of these is the lattice (fourth) isomorphism theorem. Namely, you know that the maximal ideals of a quotient ring are the maximal ideals of the form for a maximal ideal of containing . So, let's see here:

a) Yes, as you pointed out, the maximal ideals are the principal ones generated by . This follows from what I said since the maximal ideals of that contain are the principal ideals of the form where is prime.

b) Exactly. Since is a field, you know that is a PID, and since is irreducible we must have that is prime, and thus maximal. Thus, the only maximal ideals containing are the ideal itself!

c) Exactly. Since is algebraically closed, you know that all the maximal ideals are the principal ones with linear generators. Since the only linear factors dividing are your answer is correct.

d) Once again, since is a PID, the maximal ideals containing are the principal ideals generated by the irreducible factors of . In other words, the maximal ideals above are the ideals and . By the way, using this idea, you see that the maximal ideals of is just the ideal .

Best,

Alex