If $a$ and $b$ are coprime integers, is the following assertion true? $$\mathbb{Z}[i]/(a+bi)\cong\mathbb{Z}/(a^2+b^2).$$ My idea is to define $\phi:\mathbb{Z}[i]\to\mathbb{Z}/(a^2+b^2)$, $x+yi\mapsto\overline{bx-ay}$, it is obvious that $$\phi((x+yi)+(u+vi))=\phi(x+yi)+\phi(u+vi).$$ But I have difficulty with verifying that $\phi$ keeps the multiplication, i.e., $\phi((x+yi)(u+vi))=\phi(x+yi)\phi(u+vi)$ and how about the kernel of $\phi$?
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Note that your map sends $i$ to $\overline{-a}$, which is not in general a square root of $-1$. So it doesn't respect the multiplication. – Andrew Dudzik Nov 15 '21 at 09:04
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Suppose that $a$ and $b$ are coprime. – Stephen Nov 15 '21 at 09:10
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1Possible duplicate of $\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$. – Dietrich Burde Nov 15 '21 at 09:40