I have some trouble with the following problem:
Problem. Whether the ideal $I=\langle x^2+1, 5\rangle$ is a maximal ideal of $\mathbb{Z}[x]$ or not?
Solution. For this I check whether the quotient $\mathbb{Z}[x]/I$ is a field or not. So if we take $J=\langle 5\rangle$ then by 2nd isomorphism theorem we have: $$\mathbb{Z}[x]/I \cong \frac{\mathbb{Z}[x]/J}{I/J}\cong \frac{\mathbb{Z}_5[x]}{I/J}.$$
Now we have to calculate the ring $I/J=\langle x^2+1, 5\rangle/\langle 5\rangle.$ I am suspecting it to be isomorphic to $\langle x^2+1\rangle.$ To prove this I try to use 1st isomorphism theorem.
So I have to define a epimorphism from $\langle x^2+1, 5\rangle$ to $\langle x^2+1\rangle$ whose kernel is $\langle 5\rangle.$ The natural intuition should be to define the map $\phi:\langle x^2+1, 5\rangle \to \langle x^2+1\rangle$ by $(x^2+1)f(x)+5g(x) \mapsto (x^2+1)f(x).$ Then it will be a homomorphism, moreover onto. Now clearly, $5\mathbb{Z}[x] \subset \ker \phi$. Again if an element is of the form $(x^2+1)f(x)$ then it doesn't map to $0$. So $\ker \phi=\langle 5\rangle.$
Therefore by First Isomorphism theorem $\langle x^2+1, 5\rangle/\langle 5\rangle \cong \langle x^2+1\rangle.$ Then $$\mathbb{Z}[x]/\langle x^2+1, 5\rangle \cong \mathbb{Z}_5[x]/\langle x^2+1\rangle.$$
Now as the latter ring is not an field (since $x^2+1$ is not an irreducible in $\mathbb{Z}_5[x]$) so the given ideal is not an maximal ideal of $\mathbb{Z}[x]$.
Is this proof is correct? Please tell if I made any mistake. Thank you.
