A Galois conjugate of an algebraic element 
over a field 
is an element 
, where 
is a field embedding
of 
into an algebraic closure of 
that fixes every element of 
. The operation of replacing an algebraic element by one of
its Galois conjugates is sometimes called Galois conjugation.
When the minimal polynomial of 
over 
is separable, the
Galois conjugates of 
over 
are precisely the distinct roots of this minimal polynomial in an algebraic closure
of 
.
In particular, if 
is a Galois extension and 
, then the Galois conjugates of 
in 
are the elements in the group orbit 
.
For example, the Galois conjugate of 
over 
is 
, and the Galois conjugate of the golden
ratio 
over 
is
 |
Similarly, if 
is a primitive nth root of unity,
then its Galois conjugates over 
are the roots 
with gcd(k,n)=1.
