Morphism from the Gaussian integers (2)

#ring_theory

Let $i \in C$ be the usual root of unity, with $i^{2} = - 1$ , and let $Z [i] = {a + b i ∣ a, b \in Z}$ be the ring of Gaussian integers.

Prove that there exists a (nonzero) ring homomorphism $Z [i] \to Z_{5}$ .
Compute the kernel of your homomorphism explicitly, and state the conclusion given by the First Isomorphism Theorem.

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Let $i\in {\bf C}$ be the usual root of unity, with $i^2=-1$, and let ${\bf Z}[i]=\{a+bi\mid a,b\in {\bf Z}\}$ be the ring of Gaussian integers.
\begin{enumerate}[label=\alph*)]
	\item Prove that there exists a (nonzero) ring homomorphism ${\bf Z}[i]\to {\bf Z}_5$.
	\item Compute the kernel of your homomorphism explicitly, and state the conclusion given by the First Isomorphism Theorem. ({\itshape Hint:} The kernel requires two generators.)
\end{enumerate}