Math 2.0

Number theory Theorem

The integers are a Euclidean domain

Builds on Euclidean domainThe ring of integers

Statement

The ring of integers ℤ\mathbb{Z} is a Euclidean domain, with the absolute value δ(n)=∣n∣\delta(n) = |n| as Euclidean function. That is, for all a,b∈ℤa, b \in \mathbb{Z} with b≠0b \neq 0, there are q,r∈ℤq, r \in \mathbb{Z} with a=bq+ra = b q + r and 0≤r<∣b∣0 \le r < |b|.

Remarks

The proof: let rr be the least element of {a−bq:q∈ℤ}∩ℕ\{a - b q : q \in \mathbb{Z}\} \cap \mathbb{N}. Then r<∣b∣r < |b|, because otherwise r−∣b∣r - |b| would be a smaller element of this set. The pair (q,r)(q, r) with 0≤r<∣b∣0 \le r < |b| is unique. Consequences: ℤ\mathbb{Z} is a principal ideal domain, and integers factor uniquely into primes.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionThe ring of integers, example, no evidence levelThe ring of integersexampleIntegral domain, definition, no evidence levelIntegral domaindefinitionEuclidean domain, definition, no evidence levelEuclidean domaindefinitionThe integers are a Euclidean domain, statement, no evidence levelThe integers are a Eu…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. Zero divisor definition
  5. The ring of integers example
  6. Integral domain definition
  7. Euclidean domain definition
  8. The integers are a Euclidean domain statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Euclidean_domain, Examples
About this page
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Kindstatement, theorem
WritingW0 stub
MSC11A05
StatusThis copy of the site has no build data. Run m2 status 000JFQGDXG.
UsesEuclidean domain, The ring of integers

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