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
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 26$. Divisibility: Example $50$