# In the polynomial ring $\mathbb Z_3[x]$, the ideal generated by $x^6+1$ is a prime ideal.

In the polynomial ring $$\mathbb Z_3[x]$$, the ideal generated by $$x^6+1$$ is a prime ideal.

My attempt:

Theorem: Let $$F$$ be a field and $$f(x)\in F[x]$$. The ideal is $$\langle f(x) \rangle$$ is maximal iff $$f(x)$$ is irreducible over $$F$$.

Here $$Z_3$$ is a field, and the ideal generated by $$x^6+1$$ is maximal since $$x^6+1$$ is irreducible over $$\mathbb Z_3$$. So, $$\langle x^6+1 \rangle$$ is maximal and hence prime as every maximal ideal is prime ideal. Hence, the ideal generated by $$x^6+1$$ is a prime ideal.

Method 2:

Theorem:$$F[x]$$ is a PID iff $$F$$ is a field.

Theorem: Let $$R$$ be a PID. For an non-zero ideal $$I$$ such that $$I \neq R$$, then $$I$$ is prime ideal iff $$I$$ is maximal ideal. Using the above proof again, $$I=\langle x^6+1 \rangle$$ is maximal and hence prime.

But my answer key says this is false, am I doing something wrong here? Please try to explicitly point out where I am wrong and also what would be the correct approach. Thanks !

• Over the field of three elements: $x^6+1=(x^2+1)^3$. – Angina Seng Jul 9 '20 at 10:37
• Okay. This would imply it being not prime ideal, but where have I gone wrong, could you help me with that? – NAVI - s1mpleo Jul 9 '20 at 10:38
• You said $x^6+1$ is irreducible.. – justadzr Jul 9 '20 at 10:39
• Even over $\Bbb Z$, $x^6+1=(x^2+1)(x^4-x^2+1)$ is reducible. – Angina Seng Jul 9 '20 at 10:39
• Thanks @AnginaSeng and Yourong for the help. I understand my mistake now. (sorry I am unable to tag more than 1 person) – NAVI - s1mpleo Jul 9 '20 at 10:43

We know ideal $$I$$ is prime iff $$\frac{R}{I}$$ is integral domain

Here $$I= $$

We have. $$a= (x^2+1)\neq0$$ and$$b=(x^4-x^2+1) \neq 0$$ in $$\frac{Z_3[x]}{}$$

But $$ab =(x^2+1)(x^4-x^2+1) =0$$

$$\implies \frac{Z_3[x]}{}$$ is not an integral domain

$$\implies $$is not prime

• Another easy way to factor it is that by the Freshman dream theorem, $x^6+1=(x^2+1)^3$. Oh, ooop i see in the comments above it was covered as well.. – rschwieb Jul 9 '20 at 12:36