Gaussian integer
Let an agent explain Gaussian integers by definition, arithmetic, primes and applications in number theory.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent explain Gaussian integers by definition, arithmetic, primes and applications in number theory.
A complex number a + bi in which a and b are integers, forming the ring Z[i] of Gaussian integers, which has unique factorisation; Gaussian primes include 1 + i and ordinary primes congruent to 3 modulo 4, while primes congruent to 1 modulo 4 split, a fact linked to Fermat theorem on sums of two squares.
What it is for: Number theory and algebra.
It can be add, multiply and divide Gaussian integers; identify Gaussian primes; use the norm and Euclidean algorithm; connect to sums of two squares.
Distinguishing features
Integer real and imaginary parts
Forms a Euclidean domain
Unique factorisation
Norm a squared plus b squared
What it looks like
Not physical; points on the integer lattice of the complex plane.
How it is recognised
Numbers like 3 + 2i
Integer lattice points
Gaussian rationals allow fractions
Related models
is a kind of - category
is part of - field
is related to - number theory
is studied in - field
In practice
Families and kinds
Gaussian primes
units 1, -1, i and -i
associates
Gaussian integer lattice
Identifiers
Ring notation Z[i] ring of Gaussian integers
Standards and regulation
No specific regulation
Failure modes and hazards
Forgetting units when factorising
Confusing ordinary and Gaussian primes
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Arithmetic Operations.
Rules extend integers.
Operate
Add and multiply.
Operate
Operations.
- What is the product or quotient of these Gaussian integers? measurement
- Is the division exact? boundary
Norm
Norm.
Norm
Norm.
- What is the norm of this Gaussian integer? measurement
- How is the norm used in division with remainder? definition
Primes Gaussian primes.
Primes behave differently.
Test
Primality.
Test
Gaussian primality.
- Is this Gaussian integer prime? boundary
- What is its factorisation up to units? measurement
Split
Ordinary primes.
Split
Splitting of primes.
- How does an ordinary prime factor in Z[i], depending on its remainder mod 4? definition
- Why does 2 ramify? definition
Applications Uses.
Gaussian integers solve problems.
Squares
Sums of two squares.
Squares
Sums of two squares.
- How do Gaussian integers show which numbers are sums of two squares? definition
- Which theorem is involved? provenance
Euclid
Euclidean algorithm.
Euclid
Euclidean algorithm.
- How does the Euclidean algorithm find a greatest common divisor in Z[i]? action
- Why is Z[i] a Euclidean domain? definition
Learning Teaching.
Pictures help.
Visual
Lattice pictures.
Visual
Visualisation.
- How can Gaussian primes be visualised in the complex plane? action
- Which symmetries appear? definition
History
Gauss.
History
History.
- How did Gauss introduce these numbers, according to historians of mathematics? provenance
- What problem motivated them? provenance
What the second pass must settle
- Should Gaussian primes be a separate entry?
- How should other quadratic integer rings be linked?
- How should visualisations be linked?