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Let $E = \mathbb{Z}[i]/I$ where $\mathbb{Z}[i]$ denotes the ring of Gaussian integers and $I=\langle 2-i \rangle$, an ideal of $\mathbb{Z}[i]$.

We need to prove that $E$ is a field and then go on to determine the characteristic of the field.

To prove it is a field, I think the best route is to simply show that it satisfies the definition of a field. As for the characteristic, I am not really sure where to go from here. Does anyone have any suggestions for this? Thanks!

maddiemoo
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    For coprime $a,b$ we have $\mathbb Z[i]/(a - ib) \cong \mathbb Z/(a^2 + b^2).$ So we obtain the finite field $\Bbb F_5$, see the duplicate below. For a maximal ideal $I$ in $R=\Bbb Z[i]$, the quotient is always a field. – Dietrich Burde Apr 10 '22 at 19:10
  • Great, I've got that now. However, I'm struggling to prove the quotient ring is a field. A field is a commutative ring with unity and every non zero element has an inverse. Commutativity is easy to see and we have unity $1+\langle2-i\rangle$. I'm struggling to prove the inverse requirement. Any suggestions? Thanks. – maddiemoo Apr 10 '22 at 22:24
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    Yes, my suggestion is to use a different fact from abstract algebra (see above): $R/I$ is a field if and only if $I$ is maximal. Then you have the inverse for free. – Dietrich Burde Apr 11 '22 at 07:39
  • Thank you for your suggestion :) – maddiemoo Apr 11 '22 at 15:34

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