KnowraGaussian integerGaussian integerA complex number of the form a + bi, where a and b are integers and i² = −1. Gaussian integers form the ring ℤ[i].BriefConnectComplex number: A number expressible as a + bi, where a and b are real numbers and i² = −1. Gaussian integers restrict both real coordinates of complex numbers to integers.Euclidean domain: An integral domain with a size function enabling division with remainder and a Euclidean algorithm. The Gaussian norm makes ℤ[i] a Euclidean domain.Sum of two squares theorem: A characterization of integers expressible as the sum of two integer squares. The norm converts representations as sums of squares into Gaussian integer factorizations.Integer: A whole number, positive, negative, or zero. Integers sit inside ℤ[i], but some integer primes become reducible there.Ring (mathematics): An algebraic structure with addition and multiplication satisfying specified laws, including distributivity. ℤ[i] is a ring under complex addition and multiplication.Division algorithm: A procedure expressing a dividend as a divisor times a quotient plus a suitably smaller remainder. Rounding complex coordinates gives division with remainder in ℤ[i].Pythagorean triple: Three positive integers satisfying x² + y² = z². Factoring x² + y² in ℤ[i] helps construct and classify primitive triples.Rational prime: A positive integer greater than 1 with no positive divisors other than 1 and itself. Its behavior in ℤ[i] depends on its residue modulo 4.Divisibility: A relation in which one integer or ring element is an exact multiple of another. Gaussian integer factorization is defined through divisibility in ℤ[i].Euclidean algorithm: A repeated-remainder method for computing greatest common divisors in Euclidean domains. Using the Gaussian norm, it computes greatest common divisors in ℤ[i].Show all 22Linked from 15 pagesCarl 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 factorizationRelated: They retain unique factorization despite allowing factors absent from the ordinary integers.Euclid's lemmaBroader topic: Their factorization illustrates why ordinary integer primes and divisibility arguments do not transfer unchanged to every ring.Fermat's right triangle theoremRelated: Factoring sums of squares in this ring provides an algebraic route to parametrizing right triangles.Dedekind domainBroader topic: This principal ideal domain is a familiar Dedekind-domain example with unique element factorization.Friedlander–Iwaniec theoremRelated: The sum-of-squares component connects prime representation questions to arithmetic in Gaussian integers.Show all 15