Problem 5

Consider the ring $R=\mathbb{Z}[i]/\left\langle 2-i\right\rangle$.

1. Show that $5\in\left\langle 2-i\right\rangle$ and $2+\left\langle 2-i\right\rangle=i+\left\langle 2-i\right\rangle$.

2. Show that $R=\{k+\left\langle 2-i\right\rangle\mid k\in\mathbb{Z}_{5}\}$.

3. Hence prove, via an isomorphism, that $R$ is a field with order 5.