Gaußian integers

The Gaußian integers $ℤ [𝑖]$ are a $ℤ$ adjoin the imaginary unit $𝑖$ , #m/def/num hence the lattice spanned by ${1, 𝑖}$ in $ℂ$ . They form a Euclidean domain under the quadrance

N (𝑎 + 𝑏 𝑖) = (𝑎 + 𝑏 𝑖) (𝑎 - 𝑏 𝑖) = 𝑎 2 + 𝑏 2 = | 𝑎 + 𝑏 𝑖 | 2

meaning if $𝑥, 𝑦 \in ℤ [𝑖]$ with $𝑏 \neq 0$ there exist elements $𝑞, 𝑟 \in ℤ [𝑖]$ such that $𝑎 = 𝑞 𝑏 + 𝑟$ and $N (𝑟) < N (𝑦) .$

Proof of Euclidean domain

Let $𝑥, 𝑦 \in ℤ [𝑖]$ and let $𝑞$ be a lattice point such that

N (𝑥 𝑦 - 𝑞) \leq 12

Let $𝑟 = 𝑥 - 𝑦 𝑞$ . Then

N (𝑟) = N (𝑥 - 𝑦 𝑞) = N (𝑦 (𝑥 𝑦 - 𝑞)) = N (𝑦) Q (𝑥 𝑦 - 𝑞) \leq 12 N (𝑦) < N (𝑦)

as required.

Properties

The group of units is $ℤ [𝑖] \times = {1, 𝑖, - 1, - 𝑖}$

Proof of 1

Suppose $𝑎 + 𝑏 𝑖 \in ℤ [𝑖]$ is a unit, so $(𝑎 + 𝑏 𝑖) (𝑐 + 𝑑 𝑖) = 1$ for some $𝑐, 𝑑 \in ℤ$ . Then $N (𝑎 + 𝑏 𝑖) N (𝑐 + 𝑑 𝑖) = 1$ whence $N (𝑎 + 𝑏 𝑖) = 1$ so $𝑎, 𝑏 \in {1, 𝑖, - 1, - 𝑖}$ , proving ^P1.

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