Given a ring homomorphism, I'm pretty comfortable with proving whether or not it is an isomorphism. However, I'm having trouble figuring out how to systematically coming up with a ring homomorphism in the first place in order to show isomorphism. A specific example is $$ \mathbb{R}[x]/(x^2 + 1) \cong \mathbb{R}[x]/(x^2 + \alpha^2) $$ $(x^2 + 1), (x^2 + \alpha^2)$ are the principle ideals generated by $x^2 + 1, x^2 + \alpha^2$, respectively. My understanding of the (first) quotient ring in this example is the set $\{f(x) + (x^2+1) \;\vert\; f(x) = ax + b, a,b \in \mathbb{R}\}$ (please let me know if I'm wrong here).
A ring homomorphism would map the identity $1 + (x^2 + 1)$ to $1 + (x^2 + \alpha^2)$, and some $ax +b + (x^2 + 1)$ to some $a'x + b' + (x^2 + \alpha^2)$. (correct?)
This post seems to answer what I'm trying to do, but I don't understand the "abstract root" and why $\mathbb{Z}_{11}[x]/(x^2 + 1) = \mathbb{Z}_{11}[a]$ in his example, and I don't have enough rep points to comment. Could anyone show me how to do this? You don't need to use my specific example, once I understand it I'll probably able to do it myself. Thanks!