Constructing a field extension
Let
- Prove that
is irreducible. - Prove that
is a maximal ideal. - What is the cardinality of
? Justify.
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Let $f(x)=x^3+x+1\in {\bf Z}_5[x]$.
\begin{enumerate}[label=(\alph*)]
\item Prove that $f(x)$ is irreducible.
\item Prove that $\langle f(x)\rangle$ is a maximal ideal.
\item What is the cardinality of ${\bf Z}_5[x]/\langle f(x)\rangle$? Justify.
\end{enumerate}