A separable algebraic extension 
of a field 
is an algebraic extension
in which every element's extension
field minimal polynomial does not have multiple
roots. In other words, the extension
field minimal polynomial of any element is a separable
polynomial. For example,
 |
(1)
|
is a separable extension since the extension field minimal polynomial of 
, when 
, is
 |
(2)
|
In fact, in field characteristic zero, every algebraic 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 an extension
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 extension
field minimal polynomial for 
, which has one multiple
root. Since 
in characteristic 2,
 |
(5)
|
