The index of the auxiliary Kummer idele classes #
Let K be a number field containing the p-th roots of unity for a prime p. Let S and T
be disjoint finite sets of finite places of K, with S containing the places above p and
generating the ideal class group, and suppose that the S-units surject onto the local unit
p-th power classes at the places of T. This file computes the index of the image of the
auxiliary Kummer idele subgroup E_{S,T} in the idele class group C_K:
[C_K : E_{S,T} Kˣ / Kˣ] · p ^ #T = p ^ (#S + r₁ + r₂).
The principal ideles in E_{S,T} are exactly those of the (S ∪ T)-units that are p-th
powers. One inclusion is immediate. Conversely, let a ∈ Kˣ have its principal idele in
E_{S,T}. Then a is an (S ∪ T)-unit and a local p-th power at S and at the infinite
places. The surjectivity hypothesis at T therefore makes every idele class of K a norm from
the cyclic extension K(a^{1/p}), so this extension is trivial by the first fundamental
inequality, and a is a p-th power.
Writing I_{S∪T} for the (S ∪ T)-ideles and U_{S∪T} for the (S ∪ T)-units, the index of
the auxiliary idele classes is [I_{S∪T} : E_{S,T}] / [U_{S∪T} : U_{S∪T} ∩ E_{S,T}]
(index_auxiliaryKummerIdeleClassSubgroup_mul_relIndex). The numerator is p ^ (2 (#S + r₁ + r₂))
by the product formula, and the denominator is [U_{S∪T} : U_{S∪T}ᵖ] = p ^ (#S + #T + r₁ + r₂) by
Dirichlet's S-unit theorem.
For a cyclic extension L/K of degree p contained in K(U_S^{1/p}), an auxiliary set T
with p ^ #T = [K(U_S^{1/p}) : L] and E_{S,T} ⊆ N_{L/K}(C_L) exists by
exists_auxiliaryKummerIdeleClassSubgroup_le_range_ideleClassNormMap; with the index computed
here this bounds [C_K : N_{L/K}(C_L)] by p, which is the Kummer case of the second
fundamental inequality.
Main results #
TauCeti.ClassFieldTheory.unitEmbedding_mem_auxiliaryKummerIdeleSubgroup_iff: the principal ideles inE_{S,T}are those of the(S ∪ T)-units that arep-th powers.TauCeti.ClassFieldTheory.index_auxiliaryKummerIdeleClassSubgroup_mul_pow_card:[C_K : E_{S,T} Kˣ / Kˣ] · p ^ #T = p ^ (#S + r₁ + r₂).TauCeti.ClassFieldTheory.exists_auxiliaryKummerIdeleClassSubgroup_index_eq_finrank: an auxiliary subgroup of index[L : K]contained in the norm group forL ⊆ K(U_S^{1/p}).TauCeti.ClassFieldTheory.index_range_ideleClassNormMap_le_of_finrank_prime: the Kummer case of the second fundamental inequality,[C_K : N_{L/K}(C_L)] ≤ p.
References #
- J. S. Milne, Class Field Theory, v4.03, Chapter VII, §6, the proof of the second inequality.
- J. W. S. Cassels and A. Fröhlich, eds., Algebraic Number Theory, Chapter VII, §6.
The principal ideles in the auxiliary Kummer subgroup. Let S and T be disjoint, with
S containing the places above p and generating the ideal class group, and let the S-units
surject onto the local unit p-th power classes at T. Then the principal idele of a ∈ Kˣ
lies in E_{S,T} exactly when a is an (S ∪ T)-unit and a p-th power in K.
The index of the auxiliary Kummer idele classes. Let K contain the p-th roots of
unity, and let S and T be disjoint finite sets of finite places, with S containing the
places above p and generating the ideal class group. If the S-units surject onto the local
unit p-th power classes at T, then [C_K : E_{S,T} Kˣ / Kˣ] · p ^ #T = p ^ (#S + r₁ + r₂),
where r₁ + r₂ is the number of infinite places of K.
An auxiliary Kummer subgroup of index [L : K] contained in the norm group.
Here L is any intermediate field of K(U_S^{1/p}), with S containing the places above
p and generating the ideal class group.
The Kummer case of the second fundamental inequality #
The Kummer case of the second fundamental inequality. Let L/K be a Galois extension of
number fields of prime degree p, and suppose that K contains the p-th roots of unity. Then
the idele-class norm group has index at most p:
[C_K : N_{L/K}(C_L)] ≤ p.