Math 2.0

Number theory Theorem

The Gaussian integers form a Euclidean domain

Builds on Gaussian integerNorm of a Gaussian integerDivision algorithm

Statement

Let α\alpha and β\beta be Gaussian integers with β≠0\beta \neq 0. Then there are Gaussian integers qq and rr with α=qβ+r\alpha = q \beta + r and N(r)<N(β)N(r) < N(\beta), where NN is the norm. So ℤ[i]\mathbb{Z}[i], with the norm as Euclidean function, is a Euclidean domain.

Remarks

To find qq, round the real part and the imaginary part of the complex number α/β\alpha / \beta to nearest integers. This gives even N(r)≤12N(β)N(r) \le \frac{1}{2} N(\beta). As a consequence, ℤ[i]\mathbb{Z}[i] is a principal ideal domain and a unique factorization domain. The quotient and the remainder are not always unique. This is the analogue of the division algorithm for integers.

History

Carl Friedrich Gauss introduced the Gaussian integers in 1832, in his second paper on biquadratic reciprocity. In it, he proved unique factorization.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionGaussian integer, definition, no evidence levelGaussian integerdefinitionDivision algorithm, statement, no evidence levelDivision algorithmstatementNorm of a Gaussian integer, definition, no evidence levelNorm of a Gaussian in…definitionThe Gaussian integers form 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. Divisibility definition
  2. Gaussian integer definition
  3. Division algorithm statement
  4. Norm of a Gaussian integer definition
  5. The Gaussian integers form a Euclidean domain statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Gaussian_integer, Euclidean division
  2. Web https://en.wikipedia.org/wiki/Gaussian_integer, Historical background
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Kindstatement, theorem
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UsesGaussian integer, Norm of a Gaussian integer, Division algorithm

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