I need to write formula, how morphism $\pi $ is influencing general element of $K[x]/(m)$.
We have field K and polynomial $m \in K[x]$ which is irreducible over K. F is field, which is isomorphic with $K[x]/(m)$ and $\pi$ is their isomorphism.
$\pi : K[x] / (m) \rightarrow F \quad : \quad [x] \mapsto \alpha $
$\alpha$ is unknown element
I found some definition, which could help me, but I am not sure.
If $\pi$ is homomorphism of ring R to ring S, than ker $\pi$ is ideal of ring R and S is isomorphic to quotient ring $R/ker \pi$. If J is ideal of ring R, than morphism $\pi : R \rightarrow R/J$ is defined by formula $\pi(a) = a + J$ for $a \in R$ is homomorphism $R$ nad $R/J$ with ker $J$
Could someone give me a help with this?