Units of Gaussian Integers/Proof 1

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

Then $a$ and $b$ are not both $0$ as then $a + b i$ would be the zero of $\struct {\Z \sqbrk i, +, \times}$.

Then:

$\exists c, d \in \Z: \paren {a + b i} \paren {c + d i} = 1 + 0 i$

This leads (after algebra) to:

$c = \dfrac a {a^2 + b^2}, d = \dfrac {-b} {a^2 + b^2}$

Let $a^2 + b^2 = n$.

We have that $n \in \Z, n > 0$.

If $c$ and $d$ are integers, then $a$ and $b$ must both be divisible by $n$.

Let $a = n x, b = n y$.

Then:

$n^2 x^2 + n^2 y^2 = n$

and so:

$n \paren {x^2 + y^2} = 1$

Thus $n = a^2 + b^2 = 1$ and so as $a, b \in \Z$ we have:

$a^2 = 1, b^2 = 0$

or:

$a^2 = 0, b^2 = 1$

from which the result follows.

$\blacksquare$


Sources

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