The kernel of an evaluation morphism
Let
- Characterize
as a set. - Determine whether
is a maximal ideal. Fully justify your conclusion. - Determine whether
is a principal ideal. Justify by either exhibiting a generator or proving that there isn't one.
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Let ${\bf Z}[X]$ be the ring of polynmomials with integer coefficients, and let $K\subset {\bf Z}[X]$ be the kernel of the ``evaluation at $1' homomorphism
\begin{align*}
\varepsilon_1:{\bf Z}[X] &\to {\bf Z}_3\\
f(X) &\mapsto [f(1)]_3.
\end{align*}
\begin{enumerate}[label=\alph*)]
\item Characterize $K$ as a set.
\item Determine whether $K$ is a maximal ideal. Fully justify your conclusion.
\item Determine whether $K$ is a principal ideal. Justify by either exhibiting a generator or proving that there isn't one.
\end{enumerate}