Number theory MOC

Gaußian integers

The Gaußian integers ℤ[𝑖] are a Integers adjoin the imaginary unit 𝑖, num hence the lattice spanned by {1,𝑖} in Complex numbers. They form a Euclidean domain under the quadrance

N⁡(𝑎+𝑏𝑖)=(𝑎+𝑏𝑖)(𝑎−𝑏𝑖)=𝑎2+𝑏2=|𝑎+𝑏𝑖|2

meaning if 𝑥,𝑦 ∈ℤ[𝑖] with 𝑏 ≠0 there exist elements 𝑞,𝑟 ∈ℤ[𝑖] such that 𝑎 =𝑞𝑏 +𝑟 and N⁡(𝑟) <N⁡(𝑦).

Proof of Euclidean domain

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

N⁡(𝑥𝑦−𝑞)≤12

Let 𝑟 =𝑥 −𝑦𝑞. Then

N⁡(𝑟)=N⁡(𝑥−𝑦𝑞)=N⁡(𝑦(𝑥𝑦−𝑞))=N⁡(𝑦)Q⁡(𝑥𝑦−𝑞)≤12N⁡(𝑦)<N⁡(𝑦)

as required.

Properties

1. The group of units is ℤ[𝑖]× ={1,𝑖, −1, −𝑖}

Proof of 1

Suppose 𝑎 +𝑏𝑖 ∈ℤ[𝑖] is a unit, so (𝑎 +𝑏𝑖)(𝑐 +𝑑𝑖) =1 for some 𝑐,𝑑 ∈ℤ. Then N⁡(𝑎 +𝑏𝑖)N⁡(𝑐 +𝑑𝑖) =1 whence N⁡(𝑎 +𝑏𝑖) =1 so 𝑎,𝑏 ∈{1,𝑖, −1, −𝑖}, proving ^P1.

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