Given a number field 
, a Galois extension field 
, and prime ideals 
of 
and 
of 
unramified over 
, there exists a unique element 
of the Galois group 
such that for every element 
of 
,
 |
(1)
|
where 
is the norm of the prime ideal 
in 
.
The symbol 
is called an Artin symbol. If 
is an Abelian extension
of 
, the Artin symbol 
depends only on the prime ideal 
of 
lying under 
,
so it may be written as 
.
In this case, the Artin symbol can be generalized as follows. Let 
be an ideal of 
with prime factorization
 |
(2)
|
Then the Artin symbol 
is defined by
 |
(3)
|
