Identify the ring $\mathbb {Z}[i]/(2 + i) $

Question

I have to find an easy ring which this ring is isomorphic with.

I can prove that $5 = (2+i)(2-i) \in (2+i)$ but this doesn't help me further.

My book says the answer is $\mathbb {Z_5} $

Answer 1

First note that

$$\left[a+bi\right]=\left[a-2b\right]$$

in the quotient ring, meaning that you can always choose integer representatives. The next step is to find the smallest integer (in absolute value) in the ideal

$$\left<2+i\right>\ni\left(c+di\right)\left(2+i\right)=2c-d+\left(c+2d\right)i$$

This is an integer iff $c=-2d$, and then that's $-5d\in\left<2+i\right>$. The minimal case is $d=\pm1$ and you get $5\in\left<2+i\right>$. Thus in fact

$$\left[a+bi\right]=\left[a-2b\mod{5}\right]$$

and you finally conclude

$$\mathbb{Z}\left[i\right]/\left<2+i\right>\cong\mathbb{Z}_{5}$$