# Determine the maximal ideals of the rings $\mathbb{C}[X]$, $\mathbb{R}[X]/(X^2)$, $\mathbb{R}[X]/(X^2+1)$, $\mathbb{C}[X]/(X^2+1)$

I am trying to determine the maximal ideals in the following rings:
1. $\mathbb{C}[X]$
2. $\mathbb{R}[X]/(X^2)$
3. $\mathbb{R}[X]/(X^2+1)$
4. $\mathbb{C}[X]/(X^2+1)$

By reasoning as follows:
An ideal $I$ is maximal iff the quotient $R[X]/I$ is a field and by using the fact that in a PID, if $a$ is irreducible, then the ideal $(a)\triangleleft R$ is maximal.

Then I am stuck because there are many irreducibles, could anyone give me some hints of each case or some clear directions of how to proceed? Or if there are any intuitive or clever methods?

Thanks!

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While in some sense there are many irreducibles, in another sense there are only two irreducible elements in $\mathbb{R}[x]$: $x+a$ (where $a$ ranges over all real numbers) and $x^2+ax+b$ (where $a,b$ range over all real numbers such that $a^2-4b < 0$). – Hurkyl Jun 8 '13 at 21:57

1. $\mathbb C$ is a field. Thus $\mathbb C[X]$ is a Euclidean domain and every prime ideal is maximal. What are the irreducible polynomials in $\mathbb C[X]$? Use the fundamental theorem of algebra.
2. By the lattice isomorphism theorem, there is a bijective correspondence between the ideals that contain $X^2$ in $\mathbb R[X]$ and those in $\mathbb R[X]/(X^2)$. It follows that a maximal ideal in $\mathbb R[X]/(X^2)$ corresponds to a maximal ideal in $\mathbb R[X]$ that contains $(X^2)$. Find such ideal(s).
3. $X^2 + 1$ is irreducible in $\mathbb R[X]$, hence prime and maximal. What can you say about $\mathbb R[X]/(X^2+1)$?
4. Factor $X^2 + 1 = (X + i)(X - i)$ and follow the same steps as in $2$.