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Show that $\mathbb{Z}[i]/(i+4) \to \mathbb{Z}/17\mathbb{Z}$.

I know that I have to use the first Isomorphism theorem to proof this. Therefore I have to introduce a morphism $\phi : \mathbb{Z}[i] \to \mathbb{Z}/17\mathbb{Z}$ and show that the $\ker \phi = (i + 4)$. But how is the morphism defined in this case?

$\mathbb{Z}[i]$ is defined as $\mathbb{Z}[i]=\{a+b i \mid a, b \in \mathbb{Z}\}$.

Vercassivelaunos
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2 Answers2

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You can find all homomorphisms $\phi:\mathbb Z[i]\to\mathbb Z/17\mathbb Z$ and choose the one with the correct kernel. You can do so the following way: any such homomorphism can be restricted to a homomorphism $\mathbb Z\to\mathbb Z/17\mathbb Z$, of which there is only one. Now $\phi$ is completely determined by where it maps $i$. But $i$ must be mapped to an element which squares to $-1\in\mathbb Z/17\mathbb Z$. Of these there are only two: $4$ and $-4$. So any homomorphism between the given rings is of the form $a+bi\mapsto a\pm4b$. Now find their kernels.

Vercassivelaunos
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  • Also, because the goal is to show that ${\mathbb Z}[i]/(i+4)$ is isomorphic to ${\mathbb Z}/{17 \mathbb Z}$, the only candidate for $\phi(i)$ is $-4$. – Magdiragdag Apr 26 '21 at 13:19
  • Please strive not to post more (dupe) answers to dupes of FAQs. This is enforced site policy, see here. – Bill Dubuque Oct 22 '24 at 18:55
  • @BillDubuque Is this really a dupe? Sure, the answers to the linked question will apply to this "dupe" as well. But there may be answers which apply to this special case, but not the linked "original" because it is more general. A question about the proof of the FTC would not be a dupe of a question about Stokes' theorem, would it? – Vercassivelaunos Oct 23 '24 at 09:28
  • OP accepted the dupe proposed by Dietrich (that's what "Community" means in the closer list). In any case we have many tens if not hundreds of examples of these arguments by now (the site was over a decade old in 2021), so it is highly unlikely that anything new can be said. Most elementary exercises are duplicates by now. – Bill Dubuque Oct 23 '24 at 20:49
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It might be easier to look at this problem from the other direction, consider the homomorphism $\phi:\mathbb{Z}\to\mathbb{Z}[i]/(4+i)$ given by $$ n\mapsto n+(4+i) \in \mathbb{Z}[i]/(4+i) $$ What is the kernel of this map? Is it surjective?

Very much in the same spirit as this question.

Noah Solomon
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