In 'Fundamentals of Abstract Algebra' by Malik, Mordeson and Sen, at page $397$ the example no.$8$ shows that the ideal $\left<f(x)\right>$ is maximal in $\mathbb Z[x]$, where $f(x)$=$x^5$ + $12x^4$ +$9x^2$+$6$.
In solving the problem they have shown that $f(x)$ is irreducible, hence $\left<f(x)\right>$ is maximal. But we know that in a PID the an ideal is maximal iff it is generated by an irreducible element. Here $\mathbb Z[x]$ is not a PID.
Also a generalized result is if $D$ is an Integral Domain, and $a$ is a non-zero element then $a$ is irreducible iff the ideal $\left<a\right>$ is maximal among all the principal ideals of $D$.
But here the question is to show that $\left<f(x)\right>$ is maximal, so should I think that they wanted to show that $\left<f(x)\right>$ is maximal among all the principal ideals?
Hope my confusion will be cleared.
Thank you.
Best Answer
$x^5 + 12x^4 +9x^2+6$ is irreducible in $\mathbb Z[x]$ because it is irreducible mod $5$ (according to WA).
This implies that $(5,x^5 + 12x^4 +9x^2+6)$ is a maximal ideal.
In particular, $(x^5 + 12x^4 +9x^2+6)$ is not a maximal ideal.
In general,