Euclidean domain
A Euclidean domain (or Euclidean ring) is a type of ring in which the Euclidean algorithm can be used.
Formally we say that a ring 
is a Euclidean domain if:
- It is an integral domain.
- There a function
called a Norm such that for all nonzero 
there are 
such that 
and either 
or 
.
called a Norm such that for all nonzero 
there are 
such that 
and either 
or 
.Some common examples of Euclidean domains are:
- The ring of integers
with norm given by 
. - The ring of Gaussian integers
![$\mathbb Z[i]$](//latex.artofproblemsolving.com/0/d/a/0da9b2c73d35e0d39fbace6b7ff4708b92f1791b.png)
with norm given by 
. - The ring of polynomials
![$F[x]$](//latex.artofproblemsolving.com/3/0/5/3058cfec73981ccfab32d12bb576d63b2bf97bf6.png)
over any field 
with norm given by 
.
with norm given by 
.![$\mathbb Z[i]$](http://latex.artofproblemsolving.com/0/d/a/0da9b2c73d35e0d39fbace6b7ff4708b92f1791b.png)
with norm given by 
.![$F[x]$](http://latex.artofproblemsolving.com/3/0/5/3058cfec73981ccfab32d12bb576d63b2bf97bf6.png)
over any 
with norm given by 
.See also
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