Conductor of an Abelian extension
Let be an Abelian extension and let
be the corresponding subgroup of the idèle class group
(cf. Class field theory). The conductor of an Abelian extension is the greatest common divisor of all positive divisors
such that
is contained in the ray class field
.
For an Abelian extension of local fields the conductor of
is
, where
is the maximal ideal of (the ring of integers
of)
and
is the smallest integer such that
,
. (Thus, an Abelian extension is unramified if and only if its conductor is
.) The link between the local and global notion of a conductor of an Abelian extension is given by the theorem that the conductor
of an Abelian extension
of number fields is equal to
, where
is the conductor of the corresponding local extension
. Here for the infinite primes,
or
according to whether
or
.
The conductor ramification theorem of class field theory says that if is the conductor of a class field
, then
is not divisible by any prime divisor which is unramified for
and
is divisible by any prime divisor that does ramify for
.
If is the cyclic extension of a local field
with finite or algebraically closed residue field defined by a character
of degree 1 of
, then the conductor of
is equal to
, where
is the Artin conductor of the character
(cf. Conductor of a character). Here
is the separable algebraic closure of
. There is no such interpretation known for characters of higher degree.
References
[a1] | J.-P. Serre, "Local fields" , Springer (1979) (Translated from French) |
[a2] | J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, Sect. 8 |
Conductor of an Abelian extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conductor_of_an_Abelian_extension&oldid=16618