Units of Gaussian Integers/Proof 2

Proof
Let $\alpha = a + b i$ be a unit of $\struct {\Z \sqbrk i, +, \times}$.

Then by definition of unit:
 * $\exists\beta = c + d i \in \Z \sqbrk i: \alpha \beta = 1$

Let $\cmod \alpha$ denote the modulus of $\alpha$.

Then:

By Divisors of One:
 * $\cmod a^2 = 1$ or $-1$

Since $\cmod \alpha$ and $\cmod \beta$ are positive integers:
 * $\cmod \alpha^2 = a^2 + b^2 = 1$

and so either:
 * $\cmod a = 1$ and $\cmod b = 0$

or:
 * $\cmod b = 1$ and $\cmod a = 0$.

Hence the set of units of $\struct {\Z \sqbrk i, +, \times}$ is $\set {\pm 1, \pm i}$.