Kummer idele subgroups contained in norm groups #
Let L/K be a finite Galois extension of number fields whose Galois group has exponent dividing
n. If a finite set S contains every prime ramified in L, then the Kummer idele subgroup
E_S of ideles that are local n-th powers at S and at infinity, and units outside S, is
contained in the idele norm group of L/K. Consequently its image in the idele class group is
contained in the idele-class norm group. The same holds for the auxiliary subgroup E_{S,T},
whose components at a further finite set T are local units, when every place of T outside
S has a place of L above it of local degree one.
The proof is placewise. At the places of S and at infinity, exponent divisibility makes every
n-th power a local norm. At the places of T, local degree one makes every local element a
norm. Elsewhere, the extension is unramified and local units are norms.
The resulting inclusion is the norm half of the Kummer computation of the second fundamental
inequality. It converts the index of the idele-class norm group into a divisor of the index of the
Kummer idele-class subgroup.
For an intermediate field L of the Kummer field M = K(U_S^{1/p}), the Frobenius choice of an
auxiliary set T gives places of local degree one in L, so the auxiliary idele-class subgroup
E_{S,T} with p ^ #T = [M : L] is contained in N_{L/K}(C_L).
If the extension is the Kummer field of radicands that are local powers at S and at infinity,
and units outside S, then every idele class is a norm, provided S contains the places above
n and generates the class group. The places of S and infinity split completely, and the
extension is unramified elsewhere. An auxiliary set T is also allowed when the S-units
control the local power classes at T, as in Milne's Proposition 6.10.
Main statements #
auxiliaryKummerIdeleSubgroup_le_range_ideleNormMap_of_ramifiedSupport: the auxiliary Kummer idele subgroup is contained in the idele norm group.auxiliaryKummerIdeleClassSubgroup_le_range_ideleClassNormMap_of_ramifiedSupport: the corresponding inclusion after passage to idele classes.The index divisibility theorem with suffix
_of_ramifiedSupport: the resulting divisibility of indices.TauCeti.ClassFieldTheory.exists_auxiliaryKummerIdeleClassSubgroup_le_range_ideleClassNormMap: the auxiliary setTfor an intermediate Kummer field, its counting and local-power-class properties, and the inclusion of its idele-class subgroup in the norm subgroup fromL.TauCeti.ClassFieldTheory.ideleClassNormMap_kummerField_surjective: surjectivity for radicands represented by the Kummer idele subgroup.TauCeti.ClassFieldTheory.ideleClassNormMap_kummerField_surjective_of_forall_exists_mem_unit: surjectivity with an auxiliary set of places.
References #
- J. S. Milne, Class Field Theory, v4.03, Chapter VII, §§5--6, Lemmas 6.4, 6.6 and 6.9, Proposition 6.10 and §9.
- J. W. S. Cassels and A. Fröhlich, eds., Algebraic Number Theory, Chapter VII, §6.
The Kummer idele subgroup consists of norms. If the exponent of Gal(L/K) divides n
and S contains every finite place ramified in L, then every idele in the Kummer subgroup
E_{S,T} is the norm of an idele of L, provided every place of T outside S has a place of
L above it of local degree one.
The Kummer idele-class subgroup consists of norms. Under the same hypotheses as
auxiliaryKummerIdeleSubgroup_le_range_ideleNormMap_of_ramifiedSupport, every class represented
by a Kummer idele is in N_{L/K}(C_L).
The idele-class norm index divides the Kummer subgroup index. If S contains the
ramified places and n is divisible by the exponent of Gal(L/K), then
[C_K : N_{L/K}(C_L)] divides [C_K : E_{S,T}] whenever every place of T outside S has a
place of L above it of local degree one.
Idele classes are norms from a Kummer field of local powers, with an auxiliary set T.
Let S and T be disjoint finite sets of finite places of K, with S containing the places
above n and generating the ideal class group. Let Δ consist of S ∪ T-units that are n-th
powers in K_v at every v ∈ S and at every infinite place, and let m be a multiple of the
degree of M = K(Δ^{1/n}) such that the S-units surject onto ∏_{v ∈ T} U_v / U_v ^ m. Then
the norm map C_M → C_K of idele classes is surjective.
Idele classes are norms from the Kummer field of local powers. Let S be a finite set of
finite places of K containing the places above n and generating the ideal class group. If the
principal idele of every element of Δ lies in the Kummer idele subgroup E_S, that is, every
element of Δ is an S-unit that is an n-th power at the places of S and at the infinite
places, then every idele class of K is a norm from K(Δ^{1/n}).
The auxiliary Kummer idele classes are norms from an intermediate Kummer field.
Let M = K(U_S^{1/p}), where S contains the places above the prime p, and let L/K be an
intermediate field of M/K. There is a finite set T, disjoint from S, such that
p ^ #T = [M : L], the S-units surject onto the local unit p-th power classes at T, and
the image in C_K of the auxiliary Kummer subgroup E_{S,T} is contained in
N_{L/K}(C_L).