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
, 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 
, 
is the norm of the prime ideal 
in 
. 
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 
is defined by 