Show that in the ring of quaternions every non-zero element has a multiplicative inverse

Question

I've got this math question for my training to my algebra exam.

"Show that in the ring of quaternions every non-zero element has a multiplicative inverse"

The worst case is that my text book only contain a problem with quaternions but no theorems etc. How can I attach this problem and show it?

Yours,

Answer 1

Hint: Any quaternion can be written $a+bi+cj+dk$, where $a,b,c\in\mathbb{R}$.

Now try to compute $$(a+bi+cj+dk)(a-bi-cj-dk)$$

And look for something to multiply with $a+bi+cj+dk$ to get $1$.