Units of Gaussian Integers/Proof 2

Theorem

Let $\struct {\Z \sqbrk i, +, \times}$ be the ring of Gaussian integers.

The set of units of $\struct {\Z \sqbrk i, +, \times}$ is $\set {1, i, -1, -i}$.

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:

$\ds \cmod \alpha^2 \cdot \cmod \beta^2$

$=$

$\ds \cmod {\alpha \beta}^2$

$\ds $

$=$

$\ds \cmod 1^2$

$\ds $

$=$

$\ds 1$

$\cmod a^2 = 1$ 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 $\set {\pm 1, \pm i}$.

$\blacksquare$