Math 2.0

Number theory Definition

Norm on the Gaussian integers

Builds on Gaussian integers

Definition

The norm on the Gaussian integers is the map N ⁣:ℤ[i]→ℕN \colon \mathbb{Z}[i] \to \mathbb{N}, N(a+bi)=a2+b2=(a+bi)(a−bi).N(a + b i) = a^2 + b^2 = (a + b i)(a - b i).

Remarks

The norm is multiplicative: N(zw)=N(z)N(w)N(z w) = N(z) N(w) for all z,w∈ℤ[i]z, w \in \mathbb{Z}[i]. This follows from ∣zw∣=∣z∣∣w∣|z w| = |z| |w| in ℂ\mathbb{C}. N(z)=0N(z) = 0 if and only if z=0z = 0. The norm is a Euclidean function on ℤ[i]\mathbb{Z}[i]. For example, N(2+i)=5N(2 + i) = 5 and N(1+i)=2N(1 + i) = 2.

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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 integersobjectNorm on the Gaussian integers, definition, no evidence levelNorm on the Gaussian …definition
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  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
  8. Norm on the Gaussian integers definition

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3 pages build on this one.

Further reading

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