Gaussian integer
A complex number of the form , where , is calleda Gaussian integer
.
It is easy to see that the set of all Gaussian integers is a subringof ; specifically, is the smallest subring containing, whence .
is a Euclidean ring, hence a principal ring
, hence aunique factorization domain
.
There are four units (i.e. invertible elements)in the ring , namely and .Up to multiplication by units, the primes in are
- •
ordinary prime numbers
- •
elements of the form where is an ordinaryprime (see Thue’s lemma)
- •
the element .
Using the ring of Gaussian integers, it is not hard to show, for example,that the Diophantine equation has no solutions except .
Title | Gaussian integer |
Canonical name | GaussianInteger |
Date of creation | 2013-03-22 11:45:32 |
Last modified on | 2013-03-22 11:45:32 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 11 |
Author | Daume (40) |
Entry type | Definition |
Classification | msc 11R04 |
Classification | msc 55-00 |
Classification | msc 55U05 |
Classification | msc 32M10 |
Classification | msc 32C11 |
Classification | msc 14-02 |
Classification | msc 18-00 |
Related topic | EisensteinIntegers |