Evaluation at i

#ring_theory

Let $ε : R [x] \to C$ be the ring homomorphism that is evaluation at $i$ , so $ε (f) = f (i)$ . (Here $i$ denotes the complex number sometimes denoted $\sqrt{- 1}$ .)

Prove that $\ker (ε) = (x^{2} + 1) \subseteq R [x]$ .
Prove that $(x^{2} + 1)$ is a maximal ideal in $R [x]$ .

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Let $\varepsilon:{\bf R}[x]\to {\bf C}$ be the ring homomorphism that is evaluation at $i$, so $\varepsilon(f)=f(i)$. (Here $i$ denotes the complex number sometimes denoted $\sqrt{-1}$.)
\begin{enumerate}[label=\alph*)]
	\item Prove that $\ker(\varepsilon)=(x^2+1)\subseteq {\bf R}[x]$.
	\item Prove that $(x^2+1)$ is a maximal ideal in ${\bf R}[x]$.
\end{enumerate}