Math 6310 • Algebra

Prove the following assertions for the ring \(S:=\mathbb{R}[x]/(x^2+1)^2\).
\begin{enumerate}

\item
There exist exactly two ring homomorphisms \(\pi:S\to\mathbb{C}\) such that \(\pi|_{\mathbb{R}}=\operatorname{id}_{\mathbb{R}}\), the identity map on \(\mathbb{R}\).


\item
Choose one of the homomorphisms in (a), and call it \(\pi\). Prove that there exists a unique homomorphism
of rings \(\sigma:\mathbb{C}\to S\) such that \(\sigma|_{\mathbb{R}}=\operatorname{id}_{\mathbb{R}}\) and \(\pi\sigma=\operatorname{id}_{\mathbb{C}}\).


\item
Prove that \(\sigma\) extends to an isomorphism of rings \(\mathbb{C}[y]/(y^2)\to S\).

\end{enumerate}