Are these two quotient rings of $\Bbb Z[x]$ isomorphic? Are the rings $\mathbb{Z}[x]/(x^2+7)$ and $\mathbb{Z}[x]/(2x^2+7)$ isomorphic?
Attempted Solution:
My guess is that they are not isomorphic. I am having trouble demonstrating this. Any hints, as to how i should approach this?

Suppose they are isomorphic. Then
$\left(\mathbb{Z}[x]/(x^2+7)\right)/(2) \cong \left(\mathbb{Z}[x]/(2x^2+7)\right)/(2)$. Ravi helpfully pointed out that considering ideals in either ring in terms of $x$ will often give us different ideals, but we do not suffer from this problem when using ideals generated by integers such as $(2)$.
Letting overlines represent the canonical homomorphism from $\mathbb{Z}[x]$ to $\mathbb{F}_2[x]$, applying the correspondence theorem for rings on the left side tells us that $$\left(\mathbb{Z}[x]/(x^2+7)\right)/(2) \cong \left(\mathbb{Z}[x]/(2)\right)/(\overline{x^2 + 7}) \cong \mathbb{F}_2[x]/(x^2 + 1)\,.$$
Similarly, on the right, we get
$$\left(\mathbb{Z}[x]/(2x^2+7)\right)/(2) \cong \left(\mathbb{Z}[x]/(2)\right)/(\overline{2x^2 + 7}) \cong \mathbb{F}_2[x]/(1) = \{0\}\,.$$
One can verify the former ring contains four distinct elements. The latter is trivial, which is absurd.
EDIT: To further bolster intuition for an approach like this, see Bill's mention of it in his first comment on his own post.
