Units of Gaussian Integers/Proof 2

Theorem
Let $\left({\Z \left[{i}\right], +, \times}\right)$ be the ring of Gaussian integers.

The set of units of $\left({\Z \left[{i}\right], +, \times}\right)$ is $\left\{{1, i, -1, -i}\right\}$.

Proof
Let $\alpha = a + b i$ be a unit of $\left({\Z \left[{i}\right], +, \times}\right)$.

Then by definition of unit:
 * $\exists\beta = c + d i \in \Z \left[{i}\right]: \alpha \beta = 1$

Let $\left\vert{\alpha}\right\vert$ denote the modulus of $\alpha$.

Then:

Since $\left\vert{\alpha}\right\vert$ and $\left\vert{\beta}\right\vert$ are positive integers:
 * $a^2 + b^2 = \left\vert{\alpha \beta}\right\vert^2 = 1$

and so either:
 * $\left\vert{a}\right\vert = 1$ and $\left\vert{b}\right\vert = 0$

or:
 * $\left\vert{b}\right\vert = 1$ and $\left\vert{a}\right\vert = 0$.

Therefore, the set of units of $\left({\Z \left[{i}\right], +, \times}\right)$ is $\left\{ {\pm 1, \pm i}\right\}$, as required.