KnowraGaussian integerLinked fromLinked fromThe 15 pages that link to Gaussian integer, each with the reason it gives.All 15Broader topic 9Related 6Carl Friedrich GaussBroader topic: Gauss used them to extend arithmetic and clarify when integers can be represented as sums of squares.Quotient ringRelated: Quotients of this ring illustrate how ideals encode arithmetic information.Ring of integersBroader topic: They form the ring of integers of the quadratic field of rational multiples of i.Unique factorization domainBroader topic: This ring has unique factorization, though its primes differ from integer primes.Euclid's lemmaBroader topic: Their factorization illustrates why ordinary integer primes and divisibility arguments do not transfer unchanged to every ring.Dedekind domainBroader topic: This principal ideal domain is a familiar Dedekind-domain example with unique element factorization.Pythagorean tripleRelated: Factoring sums of two squares in Gaussian integers clarifies the arithmetic behind primitive triples.Algebraic integerBroader topic: This familiar ring is an example of algebraic integers inside the complex numbers.Unique factorizationRelated: They retain unique factorization despite allowing factors absent from the ordinary integers.Dirichlet's unit theoremBroader topic: Its field has no real embeddings and one complex pair, so the unit rank is zero.Fermat's theorem on sums of two squaresBroader topic: Factoring p as (a+bi)(a−bi) turns a sum of squares into a norm.Sum of two squares theoremRelated: Unique factorization in ℤ[i] explains why primes congruent to 1 modulo 4 split into two squares.Fermat's right triangle theoremRelated: Factoring sums of squares in this ring provides an algebraic route to parametrizing right triangles.Brahmagupta–Fibonacci identityBroader topic: The identity shows that products of Gaussian-integer norms remain sums of two squares.Friedlander–Iwaniec theoremRelated: The sum-of-squares component connects prime representation questions to arithmetic in Gaussian integers.