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