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Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z [ i ] {\displaystyle \mathbf {Z} } or Z [ i ] . {\displaystyle \mathbb {Z} .}

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Overview

Basic definitions

Euclidean division

Principal ideals

Gaussian primes

Unique factorization

Gaussian rationals

Greatest common divisor

Congruences and residue classes

Primitive residue class group and Euler's totient function

Historical background

Unsolved problems

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Map overview Semantic statistics

Gaussian integer

Nodes102
Edges101
Triples132
Avg. degree1.98
Density0.019608
Components1

How this topic connects Entity context

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Gaussian integer

Top relations

related to References · 44
Gaussian integer → ACM SIGPLAN Notices, Addison-Wesley, Algebra, Arithmetik, Baker, Berlin, Carl Friedrich Gauss’ Arithmetische, Comm, Commentatio, Complex Gaussian Integers, Elem, First Course In Abstract, Fraleigh, From Numbers, Gauss, Gaussian Graphics, Georg Olms Verlag, German, Göttingen, Henry
related to background · 7
Gaussian integer → Carl Friedrich Gauss, Eisenstein, Gauss, Gaussian, In, Similarly, The
related to Examples · 7
Gaussian integer → For, Gauss, Gaussian, One, There, These, Thus
related to Primitive residue class group and Euler's totient function · 7
Gaussian integer → Euler's, For Gaussian, Gaussian, Many, Obviously, The, This
related to Basic definitions · 6
Gaussian integer → Gaussian, In, It, Since, The Gaussian, When
related to Euclidean division · 6
Gaussian integer → Bézout's, Chinese, Euclid's, Euclidean, Gaussian, This
related to Principal ideals · 6
Gaussian integer → An, Euclidean, Explicitly, Gaussian, In, Since
related to Congruences and residue classes · 5
Gaussian integer → Gaussian, Given, In, The, This
related to Gaussian primes · 5
Gaussian integer → An, As, Gaussian, The, This
related to External links · 4
Gaussian integer → Conrad, Gaussian Integers, IMO Compendium, The Gaussian Integers

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Important terminology

integers gaussian norm one integer prime modulo number unique ring residue z0 thus complex form primes two ideal greatest division

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Gaussian integeris acomplex number whose real and imaginary parts are both integers0.90text
Gaussian integeris acomplex number such that its real and imaginary parts are both integers0.90text
Gaussian integeris anonnegative integer0.90text
the existence of a Euclidean algorithm for computing greatest common divisorsinstance ofand implies that Gaussian integers share with integers and polynomials many important properties0.80text
Bézout's identityinstance ofand implies that Gaussian integers share with integers and polynomials many important properties0.80text
the principal ideal propertyinstance ofand implies that Gaussian integers share with integers and polynomials many important properties0.80text
Euclid's lemmainstance ofand implies that Gaussian integers share with integers and polynomials many important properties0.80text
the unique factorization theoreminstance ofand implies that Gaussian integers share with integers and polynomials many important properties0.80text
and the Chinese remainder theoreminstance ofand implies that Gaussian integers share with integers and polynomials many important properties0.80text
all of which can be proved using only Euclidean division.A Euclidean division algorithm takesinstance ofand implies that Gaussian integers share with integers and polynomials many important properties0.80text
in the ring of Gaussian integersinstance ofand implies that Gaussian integers share with integers and polynomials many important properties0.80text
a dividend ainstance ofand implies that Gaussian integers share with integers and polynomials many important properties0.80text

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