1. list all maximal ideals

I seem to have difficulty with this type of problem - perhaps a lack of intuition about prime and maximal ideals. Here's the problem. I'm having trouble with part (d).

For each of the following rings, list all maximal ideals.
(a) $\mathbb{Z}/90\mathbb{Z}$
(b) $\mathbb{Q}[x]/(x^2+1)$
(c) $\mathbb{C}[x]/(x^2+1)$
(d) $\mathbb{Q}[x]/(x^3+x^2)$

My solutions for (a) through (c):

(a) These are generated by the prime factors of 90: (2), (3), and (5).
(b) Since $x^2+1$ is irreducible in $\mathbb{Q}[x]$, this is a field, so the only maximal ideal is (0).
(c) $\mathbb{C}[x]/(x^2+1) = \mathbb{C}[x]/((x+i)(x-i)) \cong \mathbb{C}[x]/(x+i)\times\mathbb{C}[x]/(x-i) \cong \mathbb{C}\times\mathbb{C}$, so the maximal ideals are ((0,1)) and ((1,0)), which correspond to the ideals (x+i) and (x-i) in the original ring.

I think that answer is correct, but I'm not entirely clear on how I got from the ideals in $\mathbb{C}\times\mathbb{C}$ to the ideals in the original ring.

For (d), I think it's true that $\mathbb{Q}[x]/(x^3+x^2) \cong \mathbb{Q}[x]/(x^2) \times \mathbb{Q}[x]/(x+1)$, and the second is isomorphic to $\mathbb{Q}$. But what are the maximal ideals of $\mathbb{Q}[x]/(x^2)$?

- Hollywood

2. 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 $R/I$ are the maximal ideals of the form $M/I$ for $M$ a maximal ideal of $R$ containing $I$. So, let's see here:

a) Yes, as you pointed out, the maximal ideals are the principal ones generated by $2,3,5$. This follows from what I said since the maximal ideals of $\mathbb{Z}$ that contain $(90)$ are the principal ideals of the form $(p)$ where $p\mid 90$ is prime.

b) Exactly. Since $\mathbb{Q}$ is a field, you know that $\mathbb{Q}[x]$ is a PID, and since $x^2+1$ is irreducible we must have that $(x^2+1)$ is prime, and thus maximal. Thus, the only maximal ideals containing $(x^2+1)$ are the ideal itself!

c) Exactly. Since $\mathbb{C}$ is algebraically closed, you know that all the maximal ideals are the principal ones with linear generators. Since the only linear factors dividing $x^2+1$ are $x\pm i$ your answer is correct.

d) Once again, since $\mathbb{Q}[x]$ is a PID, the maximal ideals containing $(x^3+x^2)$ are the principal ideals generated by the irreducible factors of $x^3+x^2$. In other words, the maximal ideals above $(x^3+x^2)$ are the ideals $(x+1)$ and $(x)$. By the way, using this idea, you see that the maximal ideals of $\mathbb{Q}[x]/(x^2)$ is just the ideal $(x)$.

Best,

Alex