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单词 GaussianInteger
释义

Gaussian integer


A complex numberMathworldPlanetmathPlanetmath of the form a+bi, where a,b, is calleda Gaussian integerMathworldPlanetmath.

It is easy to see that the set S of all Gaussian integers is a subringof ; specifically, S is the smallest subring containing{1,i}, whence S=[i].

[i] is a Euclidean ringMathworldPlanetmath, hence a principal ringMathworldPlanetmath, hence aunique factorization domainMathworldPlanetmath.

There are four units (i.e. invertible elements)in the ring [i], namely ±1 and ±i.Up to multiplicationPlanetmathPlanetmath by units, the primes in [i] are

  • ordinary prime numbersMathworldPlanetmath 3mod4

  • elements of the form a±bi where a2+b2 is an ordinaryprime 1mod4 (see Thue’s lemma)

  • the element 1+i.

Using the ring of Gaussian integers, it is not hard to show, for example,that the Diophantine equationMathworldPlanetmath x2+1=y3 has no solutions (x,y)×except (0,1).

TitleGaussian integer
Canonical nameGaussianInteger
Date of creation2013-03-22 11:45:32
Last modified on2013-03-22 11:45:32
OwnerDaume (40)
Last modified byDaume (40)
Numerical id11
AuthorDaume (40)
Entry typeDefinition
Classificationmsc 11R04
Classificationmsc 55-00
Classificationmsc 55U05
Classificationmsc 32M10
Classificationmsc 32C11
Classificationmsc 14-02
Classificationmsc 18-00
Related topicEisensteinIntegers
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更新时间:2025/5/4 18:05:23