Conjugate elements over a field 
are algebraic elements
of an extension field 
having the same minimal
polynomial over 
.
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 polynomial discriminant 
is negative, and hence is
their common minimal polynomial
over the field 
of real numbers.
All primitive nth roots of unity are conjugate over 
since they have the cyclotomic polynomial 
as their common minimal
polynomial. 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.
In a splitting field, the number of conjugates of an algebraic element over 
is less than or equal to the polynomial
degree of its minimal polynomial 
over 
. Equality holds iff 
has no multiple roots,
which is always the case for 
or 
. For example, the minimal
polynomial of 
over 
is
 |
(6)
|
which has four simple roots in its splitting field 
:
 |
(7)
|
These are the conjugates of 
over 
.
Conjugate elements need not be multiplicatively independent, even when their common minimal polynomial is not a cyclotomic polynomial. Schinzel's example, reported by Drmota and Skałba (1995, p. 65), is the irreducible polynomial
 |
(8)
|
Its six roots are conjugate over 
, and three distinct roots 
, 
, and 
satisfy 
.
This conjugacy relation is an equivalence relation on the set of algebraic
elements in a given extension field 
of the field 
. Every automorphism of 
fixing 
maps each equivalence class
to itself, permuting its elements.
