maximal ideals in $\mathbb{Z}_2[X]$ I am looking for maximal ideals in $\mathbb{Z}_2[X]$. I started by considering principal ideals.
\begin{eqnarray*}
\langle 0 \rangle &=& \{\}\\
\langle 1 \rangle &=& \mathbb{Z}_2[X] &=& \{0, 1\} \cup \{X, X+1\} \cup \{X^2, X^2+1, X^2+X, X^2+X+1\} \cup \dots\\
\langle x \rangle &=& x \cdot \mathbb{Z}_2[X] &=& \{0\} \cup \{X\} \cup \{X^2, X^2+X\} \cup \dots\\
\langle x+1 \rangle &=& (x+1) \cdot \mathbb{Z}_2[X] &=&  \{0\} \cup \{X+1\} \cup \{X^2+X, X^2+1\} \cup \dots&&
\end{eqnarray*}
Then I tried to combine these to get a maximal ideal, but I couldn't really see whether something like $\langle x, x+1 \rangle = x \cdot \mathbb{Z}_2[X] \cup (x+1) \cdot \mathbb{Z}_2[X]$ is already a maximal ideal or how much generators I have to add to get one.
Also, is there a better way to find all maximal ideals?
 A: Partial Answer: $\mathbb Z_2$ is a field. So the maximal ideals are corresponding to irreducible polynomials. For degrees 2 and 3, use the fact that a degree 2 or degree 3 polynomial is reducible if and only if it has a root in $\mathbb Z_2.$ As for finding all the maximal ideals, I don't know how to do it. Note that the ring $\mathbb Z_2[X]$ has infinitely many irreducible elements and hence infinitely many maximal ideals.
Added: The polynomial $x^{2^n} - x$ is separable for every $n \in \mathbb N.$  This will produce an irreducible polynomial of degree $n$ over $\mathbb Z_2$ as the corresponding splitting field will be of degree $n.$ 
A: I would say that as $\mathbf{F}_2$ is a field, $\mathbf{F}_2 [X]$ is principal, so that each (maximal) ideal is generated by one element, which is irreducible if and only if the ideal is maximal, and then I will (try...) to find all irreducible polynomials in $\mathbf{F}_2 [X]$. You can look only for monic polynomials of course. Now, for a given degree $d$, could we find at least one irreducible polynomial of degree $d$ in $\mathbf{F}_2 [X]$ ? The answer is yes, and this is how Galois constructs finite fields : $\mathbf{F}_{2^d} := \mathbf{F}_2 [X] / (P)$ where $P$ is an irreducible polynomial of degree $d$ in $\mathbf{F}_2 [X]$ - he shows that $P$ exists. Now finding such a $P$ fast is highly non-trivial, see this paper. Now, how to find all such - monic - $P$'s is another question...
