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Gaussian integer
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(Definition)
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A complex number of the form  , where
 , is called a Gaussian integer.
It is easy to see that the set  of all Gaussian integers is a subring of
 ; specifically,  is the smallest subring containing  , whence
![$ S=\mathbb{Z}[i]$](http://images.planetmath.org:8080/cache/objects/207/l2h/img7.png "$ S=\mathbb{Z}[i]$") .
![$ \mathbb{Z}[i]$](http://images.planetmath.org:8080/cache/objects/207/l2h/img8.png "$ \mathbb{Z}[i]$") is a Euclidean ring, hence a principal ring, hence a unique factorization domain.
There are four units (i.e. invertible elements) in the ring
![$ \mathbb{Z}[i]$](http://images.planetmath.org:8080/cache/objects/207/l2h/img9.png "$ \mathbb{Z}[i]$") , namely  and  . Up to multiplication by units, the primes in
![$ \mathbb{Z}[i]$](http://images.planetmath.org:8080/cache/objects/207/l2h/img12.png "$ \mathbb{Z}[i]$") are
Using the ring of Gaussian integers, it is not hard to show, for example, that the Diophantine equation  has no solutions
 except  .
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"Gaussian integer" is owned by Daume. [ full author list (2) | owner history (2) ]
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(view preamble)
Cross-references: solutions, Diophantine equation, Thue's lemma, prime numbers, primes, multiplication, ring, invertible, units, unique factorization domain, principal ring, Euclidean ring, subring, easy to see, complex number
There are 7 references to this entry.
This is version 6 of Gaussian integer, born on 2001-10-15, modified 2003-10-08.
Object id is 207, canonical name is GaussianIntegers.
Accessed 6154 times total.
Classification:
| AMS MSC: | 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers) |
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Pending Errata and Addenda
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