A separable extension 
of a field 
is one in which every element's algebraic
number minimal polynomial does not have multiple
roots. In other words, the minimal polynomial of any element is a separable
polynomial. For example,
 |
(1)
|
is a separable extension since the minimal polynomial of 
, when 
, is
 |
(2)
|
In fact, in field characteristic zero, every extension is separable, as is any finite extension of a finite
field. If all of the algebraic extensions
of a field 
are separable, then 
is called a perfect field. It is a bit more complicated
to describe a field which is not separable. Consider the field
of rational functions with coefficients in 
, infinite in size and characteristic 2 (
).
 |
(3)
|
and the extension
 |
(4)
|
For instance, 
and 
.
Then 
is not separable because 
is the minimum polynomial for 
, which has one multiple
root. Since 
in characteristic 2,
 |
(5)
|
