Constructing a field extension

#ring_theory

Let $f (x) = x^{3} + x + 1 \in Z_{5} [x]$ .
(a) Prove that $f (x)$ is irreducible.
(b) Prove that $⟨ f (x) ⟩$ is a maximal ideal.
(c) What is the cardinality of $Z_{5} [x] / ⟨ f (x) ⟩$ ? 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}