Two elements 
,
of a field 
, which is an extension field
of a field 
, are called conjugate (over 
) if they are both algebraic over 
and have the same minimal polynomial.
Two complex conjugates 
and 
(
) are also conjugate in this more abstract meaning,
since they are the roots of the following monic polynomial
 |
(1)
|
with real coefficients, which is irreducible since its discriminant 
is negative, and hence is their common minimal polynomial
over the field 
of real numbers.
All primitive 
th
roots of unity are conjugate over 
since they have the cyclotomic
polynomial 
as their common minimal polynomial. So, for, instance, the primitive fifth roots
of unity
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
are all conjugate over 
. This shows that elements (such as 
and 
) which are not conjugate over a larger field (
) may be conjugate over a smaller field.
The number of conjugates of an algebraic element over 
is less than or equal to the degree of its minimal polynomial
over 
,
and equality holds iff 
has no multiple roots in its splitting
field (which is always the case for 
or 
). For example, the minimal polynomial of 
over 
is
 |
(6)
|
which has 4 simple roots in its splitting field 
:
 |
(7)
|
These are the conjugates of 
over 
.
This conjugacy relation is an equivalence relation on the set of algebraic elements in a given extension 
of the field 
. Every element of the Galois group of the field extension
maps each conjugacy class to itself, permuting its elements.
