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In order to show that, for example $Z[i]/(2-i)\cong Z/5Z$ or $Z[i]/(4-i)\cong Z/17Z$, is there any solution that explicitly constructs a homomorphism between the two sets, establishes that it is a homomorphism, and then shows that it is both injective and surjective? The related threads on this sight invoke the first isomorphism theorem, so I was wondering if there is anything more elementary.

Can you do the same to prove the more general result that $Z[i]/(a+bi)\cong Z/N(a+bi)Z$ for $(a,b)=1$?

Tommy
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    See http://math.stackexchange.com/questions/373073/quotient-rings-of-gaussian-integers. – lhf Apr 16 '14 at 02:19
  • See this http://math.stackexchange.com/questions/385013/what-is-the-quotient-mathbb-z-sqrt3-12-sqrt3/385023#385023 – Amr Apr 16 '14 at 02:19
  • @lhf The first link uses the isomorphism theorem.. is it necessary to find the kernel? – Tommy Apr 16 '14 at 02:53
  • The computation of the kernel is feasible. If you have $n (mod\ a+bi)$ with $a,b$ coprime, then $a+bi|n$ precisely when $n=a^2+b^2,$ for the only Gaussian integers $z$ such that $z(a+bi)$ is an integer are rational multiples of $a-bi$, and since $(a,b)=1$ they are all integral multiples of $a-bi$. – Theon Alexander Jul 23 '14 at 12:23

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