Hello Forum! I have the following problem, which is related to the ring of quaternions, I tried but I had no luck in making it, I hope I can help:

Let I be the ring of integers Hamilton quaternions and define:

$\displaystyle N : I\rightarrow{}Z\ with\ N(a+bi+cj+dk)=a^2+b^2+c^2+d^2$

(The N is called norm)

Prove that an element of R is a unit if and only if it has norm +1. Addition show that $\displaystyle I^x$ is isomorphic to the quaternion group of order 8.

Thanks