I am trying to understand quotient rings and it would be really helpful if someone could show me a general way of mapping quotient rings to simpler rings.
Consider $\Bbb Z_6[x] / (x^2+5)$. I know with coefficients in $\Bbb Z_6$ we can factor
$(x^2+5) = (x-1)(x-5)$.
My guess is the answer is $\Bbb Z_6 \times \Bbb Z_6$ if we define the homomorphism $\varphi: f(x) \rightarrow (f(1),f(5)) \in \Bbb Z_6 \times \Bbb Z_6$. Am I correctly applying the first isomorphism theorem?
What about $\Bbb Z_4[x] / (x^2+1)$ which isn't reducible in $\Bbb Z_4$. I know the cosets are of the form $ax+b$. I think the quotient is a field since $(x^2+1)$ is irreducible. But I read somewhere that all finite fields have set order that is prime, and so I'm confused.
I do not have a lot of intuition here.