The auxiliary Kummer idele subgroup and its local index #
For finite sets S and T of finite places, auxiliaryKummerIdeleSubgroup S T n consists of
the ideles whose coordinates are n-th powers at S and at every infinite place, and integral
units outside S ∪ T. Thus coordinates are unrestricted at T \ S, while the power condition
still applies on S ∩ T. At exponent one it is the full group of (S ∪ T)-ideles. The index of
the exponent-n subgroup in that group is the product of the local power-class counts over S
and the infinite places. No roots-of-unity hypothesis is needed for this identity.
If the primes in S ∪ T generate the ideal class group, every idele class has an (S ∪ T)-idele
representative, unique up to the principal ideles of the (S ∪ T)-units. Consequently the index
of the image of the auxiliary Kummer subgroup in the idele class group, multiplied by the index in
the (S ∪ T)-units of those whose principal ideles lie in the auxiliary subgroup, is the same
product (index_auxiliaryKummerIdeleClassSubgroup_mul_relIndex); in particular the index divides
it.
A subgroup of the idele class group containing the n-th powers and the ray subgroup of a modulus
whose finite part is supported in S contains the image of the exponent-n subgroup for T = ∅
(auxiliaryKummerIdeleClassSubgroup_le_of_raySubgroup_le): every such idele is an n-th power
times an idele congruent to one modulo the modulus.
References #
- J. S. Milne, Class Field Theory, Chapter VII, §§5--6 and 9.
The auxiliary Kummer idele subgroup: local n-th powers at S and all infinite places,
arbitrary coordinates at T \ S, and integral units outside S ∪ T. The power condition still
applies at places in S ∩ T.
Instances For
Membership is given by the local power conditions at S and infinity and the integral-unit
condition outside S ∪ T.
At exponent one the auxiliary Kummer subgroup is the full group of (S ∪ T)-ideles.
At exponent one, membership in the auxiliary Kummer subgroup is the unit condition outside
the union S ∪ T.
The principal idele of a lies in the exponent-one auxiliary Kummer subgroup exactly when
a is an (S ∪ T)-unit.
Every auxiliary Kummer idele is an (S ∪ T)-idele.
The principal ideles lying in the (S ∪ T)-ideles are precisely the principal
(S ∪ T)-units.
The relative index of the auxiliary Kummer subgroup in the (S ∪ T)-ideles is the product
of the finite power-class counts over S and the infinite power-class counts. As usual,
Nat.card and the index are zero for an infinite quotient.
The image of the auxiliary Kummer idele subgroup in the idele class group.
Instances For
The auxiliary Kummer idele class subgroup is the image under the quotient by principal ideles.
An idele class lies in the auxiliary Kummer subgroup exactly when it has a representative in the corresponding idele subgroup.
If the classes of the primes in S ∪ T generate the ideal class group, every idele
class has an (S ∪ T)-idele representative.
If the classes of S ∪ T generate the ideal class group, the quotient map restricted
to the auxiliary Kummer subgroup of exponent one is surjective.
The index of the auxiliary Kummer idele classes. If the classes of S ∪ T generate the
ideal class group, then
[C_K : E_{S,T} Kˣ / Kˣ] · [U_{S∪T} : U_{S∪T} ∩ E_{S,T}] = [I_{S∪T} : E_{S,T}],
where I_{S∪T} is the group of (S ∪ T)-ideles, U_{S∪T} is the group of (S ∪ T)-units, and
an (S ∪ T)-unit lies in E_{S,T} when its principal idele does.
If the classes of S ∪ T generate the ideal class group, the index of the auxiliary Kummer
idele-class subgroup divides the product of the local power-class counts.
A subgroup containing the n-th powers and a ray subgroup contains a Kummer idele-class
subgroup. Let N ≤ C_K contain the n-th powers and the ray subgroup of a modulus 𝔪 whose
finite part is supported in S. Then N contains the image in C_K of the Kummer idele subgroup
E_S of ideles that are n-th powers at S and at every infinite place and units elsewhere.