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Can someone come up with an example of a maximal ideal P in $\mathbb{Z}$ such that P[X] is not maximal in $\mathbb{Z}[X]$ - the ring of polynomials with integer coefficients?

I know that the maximal ideals of $\mathbb{Z}$ are of the form $ p \mathbb{Z}$ where p is a prime number but I can't figure out the maximal ideals in $\mathbb{Z}[X]$.

Thanks!

usermath
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Katy
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    Question: "for which maximal ideals $(p)$ of $\Bbb Z$ is $(p)$ not maximal in $\Bbb Z[x]$?" Answer: all of them. For further thoughts, consider the fact $M\triangleleft R$ is maximal $\Leftrightarrow R/M$ is a field. – anon Feb 06 '14 at 20:08
  • Okay maybe i didn't formulate my question correctly - a maximal ideal P of $\mathbb{Z}$ such that P[X] is not maximal in $\mathbb{Z}[X]$ – Katy Feb 06 '14 at 20:11
  • @Katy Edit the question; don't post corrections in the comments. And the answer is still the same: every single maximal ideal of $\mathbb{Z}$ is an example. Hint: $\mathbb{Z}[X]/(p\mathbb{Z})[X]$ is not a field. – Andrew Dudzik Feb 06 '14 at 21:18

2 Answers2

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For a maximal ideal $(p)$ of ${\mathbb Z}$, the ideal $(p)$ of ${\mathbb Z}[X]$ is never maximal, since ${\mathbb Z}[X]/(p) \cong {\mathbb F}_p[X]$ which is not a field.

For a maximal ideal $(p)$ of ${\mathbb Z}$, the ideal $(p, X)$ is always a maximal ideal of ${\mathbb Z}[X]$ for very much the same reason: ${\mathbb Z}[X]/(p,X) \cong {\mathbb F}_p$, which is a field.

In general, the maximal ideals of ${\mathbb Z}[X]$ are of the form $(p,f(X))$, where $p \in {\mathbb Z}$ is prime and $f(X) \in {\mathbb Z}[X]$ is irreducible modulo $p$.

Magdiragdag
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$\begin{eqnarray}\rm {\bf Hint}\ \ \ Notice\ that\ the \ ideals \ \ (p) &\subsetneq&\rm (p,x)&\subsetneq&\rm (1)\ \ are\ distinct\ \ \,primes\ \ [or\ (1)]\\ \rm because\ their\ residue\ rings\ \ \ \Bbb F_p[x] &\supsetneq&\ \ \, \Bbb F_p&\supsetneq&\rm (0)\ \ are\ distinct\ domains\ [or\ (0)]\end{eqnarray}$

Bill Dubuque
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