Number theory MOC

Gaußian integers

The Gaußian integers are a adjoin the imaginary unit , #m/def/num hence the lattice spanned by in . They form a Euclidean domain under the quadrance

meaning if with there exist elements such that and

Proof of Euclidean domain

Let and let be a lattice point such that

Let . Then

as required.

Properties

  1. The group of units is
Proof of 1

Suppose is a unit, so for some . Then whence so , proving ^P1.

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