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Let $R=\mathbb{Z}[i] := \{a + ib \in \mathbb{ C} : a, b \in \mathbb{Z}\}$ be the ring of Gaussian integers. Let $I := (4 + 5i)$ be the principal ideal generated by $4 + 5i$ and $S := \mathbb{Z}[i]/ I $ be the quotient ring. Then

  • $S$ is an integral domain.
  • $S$ is not an integral domain.
  • $S$ is a field.
  • $S$ is an integral domain but not a field

I think this will be integral domain because $\Bbb{Z}$ is integral domain but not field.

Any hints/ solution

Thanks u

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  • $\begingroup$ $\Bbb Z[i]/(4+5i)\cong \Bbb Z / 41 \Bbb Z$, so $S$ is a field $\endgroup$ – Mustafa Oct 9 '18 at 4:10
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$4+5i$ is irreducible in $\Bbb{Z}[i]$, since $\vert \vert 4+5i \vert \vert=4^2+5^2=41$, a prime number

So $\langle 4+5i \rangle$ is maximal and $R/I$ is a field as well as an integral domain

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