Math 2.0

Number theory Algebraic number theory Object

Gaussian integers

Builds on SubringIntegral domain

Description

The ring of Gaussian integers is ℤ[i]={a+bi:a,b∈ℤ},\mathbb{Z}[i] = \{a + b i : a, b \in \mathbb{Z}\}, where i∈ℂi \in \mathbb{C} satisfies i2=−1i^2 = -1. It is a subring of ℂ\mathbb{C}, with (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a + b i)(c + d i) = (a c - b d) + (a d + b c) i. As a subring of a field, it is an integral domain.

Remarks

The norm N(a+bi)=a2+b2N(a + b i) = a^2 + b^2 is multiplicative. The [[units of ℤ[i]\mathbb{Z}[i]|06RH4XAGRY]] are 1,−1,i,−i1, -1, i, -i. The ring ℤ[i]\mathbb{Z}[i] is a Euclidean domain, so it has unique factorization. A prime number pp is a sum of two squares if and only if pp is not prime in ℤ[i]\mathbb{Z}[i]. This holds exactly for p=2p = 2 and the primes p≡1(mod4)p \equiv 1 \pmod 4.

Reading path

To understand this page from the ground up, read these 7 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 domaindefinitionGaussian integers, object, no evidence levelGaussian integersobject
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. Gaussian integers object

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Gaussian_integer, Basic definitions
About this page
Tag0K5W69AC0J
URIm2://0K5W69AC0J
Kindobject
WritingW1 stated
MSC11R11
Instance ofEuclidean domain
UsesSubring, Integral domain
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What this page needs for W2

  • section Invariants: a table of the main invariants (0 of 1)

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