Polynomial irreducible - maximal ideal I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal.
$I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$
$\mathbf 1)$ Is $I_1$ a maximal ideal in $\mathbb{Q}[x]$?
Yes, since $I_1$ is irreducible with $p=3$ using Eisenstein's criterion, thus maximal ideal.
$\mathbf 2)$ Is $I_2$ a prime ideal in $\mathbb{Q}[x]$?
Yes, since $I_2$ is obviously irreducible, and thus a maximal ideal, and every maximal ideal is a prime ideal.
$\mathbf 3)$ Is $I_2$ a maximal ideal in $\mathbb{Z}[x]$?
Yes, $I_2$ is obviously irreducible, and thus a maximal ideal.
$\mathbf{Edit:}$ No, as it is not a field.
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Am I right in my conclusions?
Appreciate any help.
 A: There is a theorem that you can use: Given a commutative ring $R$ with identity, $I$ is a maximal ideal in $R$ if and only if $R/I$ is a field. (Similarly, $I$ is a prime ideal if and only if $R/I$ is an integral domain.)
What do elements in $\mathbb{Z}[x]/\langle x - 1\rangle$ look like? Do they form a field?
A: In 3, your argument is wrong. In an integral domain, $a$ is irreducible iff $(a)$ is maximal among principal ideals, but $\mathbb{Z}[x]$ is not a PID, thus you cannot conclude that $(a)$ is maximal. In fact it is not, because $\mathbb{Z}[x]/\langle x-1\rangle$ is not a field. So for example $\langle 2,x-1\rangle$ is a maximal ideal containing $\langle x-1\rangle$.
A: In a commutative ring $R$ with $1$ and $a\ne 0$
\begin{array}\
R/(a) \text{ integral domain} &\iff &(a) \text{ prime ideal} & \iff & a \text{ prime}\\
&&&  & & \\
\Uparrow&&(a) \text{ maximal among principal} & \Longleftarrow & a \text{ irreducible} &\\
 && &  &  & \\
R/(a) \text{ is a field} & \iff &(a) \text{ maximal ideal} 
\end{array}
In an integral domain $R$ and $a\ne 0$
\begin{array}\
R/(a) \text{ integral domain} &\iff &(a) \text{ prime ideal} & \iff & a \text{ prime}\\
&&&  & \Downarrow & \\
\Uparrow&&(a) \text{ maximal among principal} & \iff & a \text{ irreducible} &\\
 && &  &  & \\
R/(a) \text{ is a field} & \iff &(a) \text{ maximal ideal} 
\end{array}
In a UFD $R$ and $a\ne 0$
\begin{array}\
R/(a) \text{ integral domain} &\iff &(a) \text{ prime ideal} & \iff & a \text{ prime}\\
&&&  & \Updownarrow & \\
\Uparrow &&(a) \text{ maximal among principal} & \iff & a \text{ irreducible} &\\
 && &  &  & \\
R/(a) \text{ is a field} & \iff &(a) \text{ maximal ideal} 
\end{array}
In a PID $R$ and $a\ne 0$
\begin{array}\
R/(a) \text{ integral domain} &\iff &(a) \text{ prime ideal} & \iff & a \text{ prime}\\
&& &  & \Updownarrow & \\
\Uparrow &&(a) \text{ maximal among principal} & \iff & a \text{ irreducible} &\\
 &&\Downarrow &  &  & \\
R/(a) \text{ is a field} & \iff &(a) \text{ maximal ideal} 
\end{array}
