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I want to find the maximal ideals of the ring $\mathbb{Z}[x]$ containing $30$ and $x^2+1$.

Any such ideal will contain the ideal $(30, x^2+1)$, so we are searching for maximal ideals in the ring $$\mathbb{Z}[x] / (30,x^2+1) \cong \mathbb{Z}_{30}[x] /(x^2+1) \cong \mathbb{Z}_5[x] /(x^2+1) \oplus\mathbb{Z}_3[x] /(x^2+1) \oplus \mathbb{Z}_2[x] /(x^2+1)$$

Now, we search for the ideals in the summands. Factorizing, $x^2 + 1 = (x+3)(x+2) \bmod 5$, $x^2 + 1$ is irreducible $\bmod 3$, and $x^2 + 1 = (x+1)^2 \bmod 2$.

From this, we see that $\mathbb{Z}_3[x] /(x^2+1)$ is a field - there are no nonzero ideals. Moreover, we get ideals corresponding to the factored components in the other summands.

How do I find the maximal ideals in $\mathbb{Z}_{30}[x] /(x^2+1)$ from here? Furthermore, how do I find the generators of the corresponding ideals in $\mathbb{Z}_{30}$?

user26857
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  • This shouldn't be that hard. For instance, you want the corresponding maximal ideal to $(x+3)\bmod 5$. This is simply $(5,x+3)$. For $(x+1)\bmod 2$ consider $(2,x+1)$, and so on. – user26857 Oct 05 '17 at 06:12
  • Since an ideal is prime iff the quotient is a domain, and a direct sum of rings is never a domain, you get that the primes of $A \oplus B$ are the ideals $(A,p)$ and $(q,B)$ where $p \subset B$ and $q \subset A$ are primes. Same for maximal ideals. So you've already found the maximal ideals on the right; to get them back on the left, remember what map is used in the Chinese remainder theorem. – SdV May 12 '21 at 12:29
  • This question is more or less contained in this older question. – Jyrki Lahtonen Jul 11 '24 at 19:49

1 Answers1

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Using the $\mathbb Z$-algebra isomorphism $\mathbb Z[x]/\langle x^2+1\rangle\cong \mathbb Z[i]$, you can conclude that $\mathbb Z[x]/\langle x^2+1,30\rangle \cong \mathbb Z[i]/\langle 30\rangle $. Now $\mathbb Z[i]$ is a PID. So the maximal ideals of $\mathbb Z[i]/\langle 30\rangle $ correspond to prime factors of $30$ in the UFD $\mathbb Z[i]$. We have the prime factorization $30=(1+i).(1-i).3.(2+i).(2-i)$. So the required maximal ideals in the ring $ \mathbb Z[i]/\langle 30\rangle $ are $\langle 1\pm i\rangle$, $\langle 3\rangle$ and $\langle 2\pm i\rangle$. These correspond to the ideals $\langle1\pm \bar x \rangle$, $\langle 3 \rangle$ and $\langle 2\pm \bar x\rangle$ in the ring $\mathbb Z[x]/\langle x^2+1,30\rangle$. So the required maximal ideals are $\langle2,1\pm x \rangle$, $\langle 3, x^2+1 \rangle$ and $\langle 5,2\pm x\rangle$.

user6
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