Math 2.0

statement proposition

Units of the Gaussian integers

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.

Explanations

L1 human

A unit must have norm 11, so it lies on the unit circle. The only points with integer coordinates on the unit circle are 11, ii, −1-1 and −i-i.

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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
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Gaussian_integer, Basic definitions

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