Units of Gaussian Integers/Proof 2
Jump to navigation
Jump to search
Theorem
Let $\struct {\Z \sqbrk i, +, \times}$ be the ring of Gaussian integers.
The set of units of $\struct {\Z \sqbrk i, +, \times}$ is $\set {1, i, -1, -i}$.
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:
\(\ds \cmod \alpha^2 \cdot \cmod \beta^2\) | \(=\) | \(\ds \cmod {\alpha \beta}^2\) | Modulus of Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod 1^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
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 $\set {\pm 1, \pm i}$.
$\blacksquare$