First, I am sorry to ask this question because I cannot find the detail of the proof of the theorem.
Consider two quotient rings $ \mathbb Z [x]/(x^2+ax+b) $ and $ \mathbb Z [x]/(x^2+cx+d) $ We know that two rings are isomorphic if they have the same discriminant. What I am not sure about is how to construct this isomorphism. If these two polynomials do not have roots in $\mathbb Z$, I believe the isomorphism will be the one by sending one root of one polynomial because we can write these two rings as $\mathbb Z[x]$ adjoining by root.
I would appreciate it if someone can give me a reference of the proof of the theorem and tell me whether the isomorphism I am thinking about is correct.