Math 2.0

Number theory Proposition

Units of the Gaussian integers

Builds on Gaussian integersNorm on the Gaussian integersUnit of a ring

Statement

The units of the Gaussian integers ℤ[i]\mathbb{Z}[i] are 11, −1-1, ii and −i-i.

Remarks

The proof uses the norm: if uv=1u v = 1, then N(u)N(v)=1N(u) N(v) = 1, so N(u)=1N(u) = 1. The only solutions of a2+b2=1a^2 + b^2 = 1 in integers are (±1,0)(\pm 1, 0) and (0,±1)(0, \pm 1). Conversely, an element of norm 11 is a unit, because (a+bi)(a−bi)=1(a + b i)(a - b i) = 1. The group of units is cyclic of order 44, generated by ii.

Reading path

To understand this page from the ground up, read these 10 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 ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionGaussian integers, object, no evidence levelGaussian integersobjectNorm on the Gaussian integers, definition, no evidence levelNorm on the Gaussian …definitionUnits of the Gaussian integers, statement, no evidence levelUnits of the Gaussian…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. Unit of a ring definition
  6. Zero divisor definition
  7. Integral domain definition
  8. Gaussian integers object
  9. Norm on the Gaussian integers definition
  10. Units of the Gaussian integers statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Gaussian_integer, Basic definitions
About this page
Tag06RH4XAGRY
URIm2://06RH4XAGRY
Kindstatement, proposition
WritingW0 stub
MSC11R11
StatusThis copy of the site has no build data. Run m2 status 06RH4XAGRY.
UsesGaussian integers, Norm on the Gaussian integers, Unit of a ring

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.