Is $(X^2-2,7)$ a prime ideal or maximal ideal in $\mathbb{Z}[X]$? Using the third isomorphism theorem, we find
$$
\Big(\mathbb{Z}[X]/ (X^2-2)\Big)/\Big( (X^2-2,7)/(X^2-2)\Big) \cong \Big(\mathbb{Z}[X]/ (X^2-2)\Big)/(7) 
$$
and
$$
 \Big(\mathbb{Z}[X]/ (X^2-2)\Big)/(7)  \cong (\mathbb{Z}/7\mathbb{Z})[\sqrt2],
$$
but this doesn't really help. Any insights or hints? 

I mean, you are basically there. You just thought about the last part wrong.
As you noticed
$$\mathbb{Z}[x]/(7,x^2-2)\cong (\mathbb{Z}/7\mathbb{Z})[x]/(x^2-2)$$
Since $\mathbb{Z}/7\mathbb{Z}$ is a field, to verify the last quotient is a field/domain you must only prove that $x^2-2$ is irreducible. But, degree two polys are irreducible over a field if and only if they have no root in the field. Does $x^2-2$ have a root in $\mathbb{Z}/7\mathbb{Z}$?
