Morphism from the Gaussian integers
Let
- Define a ring homomorphism from
. You must prove it is a ring homomorphism. - Find, with proof, a generator for the kernel of your ring homomorphism.
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Let $i$ be the imaginary number, let ${\bf Z}[i]=\{a+bi\,\mid \, a,b\in {\bf Z}\}$, a principal ideal domain, and let ${\bf Z}_2$ be the finite ring of integers modulo 2.
\begin{enumerate}[label=\alph*)]
\item Define a ring homomorphism from ${\bf Z}[i]\to {\bf Z}_2$. You must prove it is a ring homomorphism.
\item Find, with proof, a generator for the kernel of your ring homomorphism.
\end{enumerate}