An evaluation morphism

#ring_theory

Let $z \in C$ be a complex number and let $ϵ_{z} : R [x] \to C$ be the evaluation homomorphism given by $ϵ_{z} (p) = p (z)$ for each $p \in R [x]$ .

Show that $\ker (ϵ_{z})$ is a prime ideal.
Compute $\ker (ϵ_{1 + i}), im (ϵ_{1 + i})$ and then state the conclusion of the First Isomorphism Theorem applied to the homomorphism $ϵ_{1 + i}$ .

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Let $z\in {\bf C}$ be a complex number and let $\epsilon_z:{\bf R}[x]\to {\bf C}$ be the evaluation homomorphism given by $\epsilon_z(p)=p(z)$ for each $p\in {\bf R}[x]$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Show that $\ker(\epsilon_z)$ is a prime ideal.
	\item Compute $\ker(\epsilon_{1+i}), \operatorname{im}(\epsilon_{1+i})$ and then state the conclusion of the First Isomorphism Theorem applied to the homomorphism $\epsilon_{1+i}$.
\end{enumerate}