0

Let $I=\langle 3, f\rangle$ be an ideal in $\mathbb{Z}[X]$, where $f(x)=x^3+x^2+1$. How do I show that $I$ is a prime ideal?

I know that $I$ is not a principal ideal, using an argument similar to: Why is $(3,x^3-x^2+2x-1)$ not principal in $\mathbb{Z}[x]$?

I tried proving using definition, letting $ab\in I$ and trying to show that $a\in I$ or $b\in I$.

Let $ab\in I$. Then $ab=3g+fh$ for some $g,h\in\mathbb{Z}[X]$.

At this point, I have no idea what to do. I tried using the division algorithm to write $g=q_1f+r_1$, where $\deg r_1<3$, and $h=q_2f+r_2$, where $\deg r_2<3$.

Then $ab=3q_1f+3r_1+f^2q_2+fr_2$. I am not sure if this is the right track, or it just complicates things further.

Thanks for any help!

I also had the idea of using: $I$ is a prime ideal iff $R/I$ is an integral domain, but I realised that it amounts to the same approach as above.

Community
  • 1
yoyostein
  • 19,608
  • 1
    Try showing the contrapositive: that for $a, b \in \mathbb{Z}[X]$, if $a, b \notin I$, then $ab \notin I$. Consider the remainders upon reducing $a$ and $b$ in terms of $3$ and $f$. – Michael L. Nov 05 '16 at 08:12

1 Answers1

4

It's not a prime ideal. $(x-1)(x^2-x-1)\in I$, while $x-1\not\in I$ and $x^2-x-1\not\in I$.

Cave Johnson
  • 4,270
  • 1
    Isn't it easier by noting that $f(1)=3$, so $f(x)-3=(x-1)(x^2+2x+2)\in I$? – egreg Nov 05 '16 at 09:44
  • @egreg Thanks, you're quite right :) BTW, I'm now curious about the examples where $I$ is a prime ideal in $\mathbb{Z}[x]$ but not a principal ideal. Since it can be shown that if a prime ideal $\mathfrak p$ satisfies $\mathfrak{p}\cap\mathbb{Z}={0}$, then $\mathfrak p$ is a principal ideal, so the example must be of the form $(n)+(f)$, where $n$ is a composite number. Do you have any idea? – Cave Johnson Nov 05 '16 at 09:52
  • If $\mathfrak{p}$ is prime and $\mathfrak{p}\cap\mathbb{Z}=n\mathbb{Z}$, with $n>0$, then $n$ must be prime. – egreg Nov 05 '16 at 10:02
  • @egreg I see. So there is actually no prime ideal in $\mathbb{Z}[x]$ which is not a principal ideal? – Cave Johnson Nov 05 '16 at 10:04
  • 1
    Why? $(2,x)$ is prime and not principal. – egreg Nov 05 '16 at 10:06
  • @egreg Unfortunately I mixed up the principal ideal and maximal ideal... Thanks a lot for clarification! – Cave Johnson Nov 05 '16 at 10:09
  • Thanks a lot. What is the fastest way to see $(x-1), (x^2+2x+2)$ not in $I$? – yoyostein Nov 05 '16 at 16:34