Math 2.0

Number theory Theorem

The Gaussian integers are a Euclidean domain

Builds on Gaussian integersNorm on the Gaussian integersEuclidean domain

Statement

The ring of Gaussian integers ℤ[i]\mathbb{Z}[i] is a Euclidean domain, with the norm N(a+bi)=a2+b2N(a + b i) = a^2 + b^2 as Euclidean function. That is, for all z,w∈ℤ[i]z, w \in \mathbb{Z}[i] with w≠0w \neq 0, there are q,r∈ℤ[i]q, r \in \mathbb{Z}[i] with z=qw+rz = q w + r and N(r)<N(w)N(r) < N(w).

Remarks

The proof: write z/w=x+yiz / w = x + y i in ℂ\mathbb{C}. Let q=m+niq = m + n i, where mm and nn are integers nearest to xx and yy. Then ∣z/w−q∣2≤14+14=12|z/w - q|^2 \le \tfrac{1}{4} + \tfrac{1}{4} = \tfrac{1}{2}, so r=z−qwr = z - q w has N(r)≤12N(w)<N(w)N(r) \le \tfrac{1}{2} N(w) < N(w). A consequence is that ℤ[i]\mathbb{Z}[i] is a principal ideal domain and a unique factorization domain.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionSubring, definition, no evidence levelSubringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionEuclidean domain, definition, no evidence levelEuclidean domaindefinitionGaussian integers, object, no evidence levelGaussian integersobjectNorm on the Gaussian integers, definition, no evidence levelNorm on the Gaussian …definitionThe Gaussian integers are a Euclidean domain, statement, no evidence levelThe Gaussian integers…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Subring definition
  5. Zero divisor definition
  6. Integral domain definition
  7. Euclidean domain definition
  8. Gaussian integers object
  9. Norm on the Gaussian integers definition
  10. The Gaussian integers are a Euclidean domain statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Gaussian_integer, Euclidean division
About this page
Tag0PN13BB7BT
URIm2://0PN13BB7BT
Kindstatement, theorem
WritingW0 stub
MSC11R11
StatusThis copy of the site has no build data. Run m2 status 0PN13BB7BT.
UsesGaussian integers, Norm on the Gaussian integers, Euclidean domain

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