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## Homework Statement

Let ##S=\{a+bi+cj+dk \mid a,b,c,d \in \mathbb{Z}\}## be the ring of integral Hamiltonian quaternions, where multiplication is defined using the same rules as in ##\mathbb{H}##, the ring of real Hamiltonian quaternions. Define a function $$N:S\to\mathbb{Z}, N(a+bi+cj+dk)=a^2+b^2+c^2+d^2.$$

Show that if ##N(\alpha)=1##, then ##\alpha## is a unit

## Homework Equations

## The Attempt at a Solution

So suppose we know that ##N(\alpha) = \alpha\bar{\alpha}##, where the bar is the conjugate of the quaternion. If ##N(\alpha)=1##, then ##\alpha\bar{\alpha} = 1##. So to show that ##\alpha## is a unit, all I need to show is the other direction, that ##\bar{\alpha}\alpha=1##. How can I do this? In groups one implies the other since we have inverses, but this is not the case with rings.