The n-th power subgroup of a local field #
Let K be a nonarchimedean local field. This file counts the power classes Kˣ ⧸ (Kˣ)ⁿ, where
(Kˣ)ⁿ is the range of powMonoidHom n : Kˣ →* Kˣ, and studies (Kˣ)ⁿ further when n is
invertible in 𝒪[K], that is, prime to the residue characteristic.
For every n with (n : K) ≠ 0,
#(Kˣ ⧸ (Kˣ)ⁿ) = n · #μ_n(K) · q ^ v_K(n),
where μ_n(K) is the group of n-th roots of unity in K, q is the cardinality of the residue
field and v_K(n) is the normalized valuation natCastValuation K n hn. In characteristic zero,
for instance for a finite extension of ℚ_[p], this covers every n ≠ 0, including multiples of
the residue characteristic. The proof compares the index of the n-th powers with the number of
n-torsion elements, which is unchanged on passing to a subgroup of finite index
(Subgroup.index_range_pow_mul_card_ker). The general reduction to principal units splits
Kˣ ≅ ℤ × μ_{q-1}(K) × U(K,1), where the factor ℤ contributes n. Inside U(K,1), the
deep subgroup U(K, v_K(n) + 1) has no n-torsion and is carried by the n-th power map onto
U(K, 2 v_K(n) + 1), of index q ^ v_K(n) (TauCeti.map_powMonoidHom_unitFiltration).
In particular there are 4 square classes when 2 is a unit of 𝒪[K].
When n is invertible in 𝒪[K], so that v_K(n) = 0, the n-th power map is an automorphism of
each positive-depth step U(K,i+1), so every principal unit is an n-th power. Hence (Kˣ)ⁿ
contains an open subgroup and is open, and hence closed, in Kˣ. Openness does not follow from
any finiteness of the quotient Kˣ ⧸ (Kˣ)ⁿ: a subgroup of finite index in a topological group need
not be open. Since the principal units are exactly the units of 𝒪[K] that reduce to 1, a unit
of 𝒪[K] is then an n-th power in K precisely when its residue is an n-th power in 𝓀[K].
At n = 2 that is the criterion for a unit of 𝒪[K] to be a square. As a consequence, a subgroup
of Kˣ is open as soon as the exponent (for instance, the index) of the quotient by it is
invertible in 𝒪[K], since it then contains the power subgroup attached to that exponent.
The openness results are stated in PowerSubgroup.Open, including the general case
(n : K) ≠ 0 obtained from powers of deep units.
Main results #
TauCeti.card_powerClasses:#(Kˣ ⧸ (Kˣ)ⁿ) = n · #μ_n(K) · q ^ v_K(n)for(n : K) ≠ 0.TauCeti.card_powerClasses_eq_mul_inv_normalizedAbsoluteValue: the same count as an equality#(Kˣ ⧸ (Kˣ)ⁿ) = n · #μ_n(K) · ‖n‖_K⁻¹inℚ≥0, with‖·‖_Kthe normalized absolute value.TauCeti.finiteIndex_range_powMonoidHom:(Kˣ)ⁿhas finite index inKˣfor(n : K) ≠ 0.TauCeti.map_powMonoidHom_unitFiltration_succ_of_isUnit: then-th power map carriesU(K,i+1)onto itself.TauCeti.unitFiltration_one_le_range_powMonoidHom_of_isUnit: every principal unit is ann-th power,U(K,1) ≤ (Kˣ)ⁿ.TauCeti.unitsMap_subtype_mem_range_powMonoidHom_iffandTauCeti.isSquare_unitsMap_subtype_iff: a unit of𝒪[K]is ann-th power, respectively a square, inKexactly when its residue is one in𝓀[K].TauCeti.disjoint_rootsOfUnity_unitFiltration_one_of_isUnit: no nontrivialn-th root of unity is a principal unit.TauCeti.powMonoidHom_unitFiltration_succ_bijective_of_isUnit: then-th power map is a bijection ofU(K,i+1).TauCeti.card_powerClasses_eq_of_index_unitFiltration_one: the power-class count reduces to the index/kernel ratio of the power map on the principal units.TauCeti.card_powerClasses_of_isUnit:#(Kˣ ⧸ (Kˣ)ⁿ) = n · #μ_n(K).TauCeti.finiteIndex_range_powMonoidHom_of_isUnit:(Kˣ)ⁿhas finite index inKˣ.TauCeti.card_squareClasses_of_isUnit:#(Kˣ ⧸ (Kˣ)²) = 4when2is a unit of𝒪[K].TauCeti.card_squareClasses:#(Kˣ ⧸ (Kˣ)²) = 4 · q ^ v_K(2)when2 ≠ 0inK.
Implementation notes #
In the theorems assuming IsUnit (n : 𝒪[K]), this hypothesis already forces n ≠ 0, so
no separate nonvanishing assumption is taken. The general reduction theorem
card_powerClasses_eq_of_index_unitFiltration_one instead requires n ≠ 0 explicitly. In mixed
characteristic the same openness holds for every n ≠ 0, using the binomial power identity on
deep units instead of Hensel's lemma at 1, and in equal characteristic p the range
of powMonoidHom p is not open. Likewise the count acquires the factor q ^ v_K(n) when n is not
a unit, and in equal characteristic p the quotient Kˣ ⧸ (Kˣ)ᵖ is infinite.
References #
- J.-P. Serre, Corps Locaux, Chapter V, §3.
- J. Neukirch, J. Schmidt, K. Wingberg, Cohomology of Number Fields, Chapter VII, §3.
- J. Neukirch, Algebraic Number Theory, Chapter II, §5.
For n invertible in 𝒪[K], the n-th power map carries each positive-depth step
U(K,i+1) of the unit filtration onto itself. This is the case v_K(n) = 0 of
map_powMonoidHom_unitFiltration.
For n invertible in 𝒪[K], every principal unit of K is an n-th power.
n-th powers away from the residue characteristic are detected in the residue field.
For n invertible in 𝒪[K], a unit of 𝒪[K] is an n-th power in K exactly when its residue
is an n-th power in 𝓀[K].
Away from residue characteristic two, a unit of 𝒪[K] is a square in K exactly when its
residue is a square in 𝓀[K].
For n invertible in 𝒪[K], the only n-th root of unity in K that is a principal unit
is 1: the groups μ_n(K) and U(K,1) intersect trivially. This is the case v_K(n) = 0 of
disjoint_rootsOfUnity_unitFiltration.
For n invertible in 𝒪[K], the n-th power map is a bijection of each positive-depth step
U(K,i+1) of the unit filtration.
Reduction of the power-class count to the principal units. Suppose the n-th power map
on U(K,1) has index #U(K,1)[n] · c. Then
#(Kˣ/(Kˣ)ⁿ) = n · #μ_n(K) · c.
The factor n comes from the normalized valuation Kˣ → ℤ; the prime-to-residue-
characteristic roots of unity and the principal units together account for all of μ_n(K).
Thus the remaining local-field input to a power-class formula is exactly the index/kernel ratio
of the power map on the principal units.
The number of n-th power classes. For n with (n : K) ≠ 0, the quotient
Kˣ ⧸ (Kˣ)ⁿ has n · #μ_n(K) · q ^ v_K(n) elements, where μ_n(K) is the group of n-th roots
of unity in K, q is the cardinality of the residue field and v_K(n) is the normalized
valuation of n. This holds in either characteristic; in characteristic zero, for instance for
K a finite extension of ℚ_[p], it applies to every n ≠ 0, including multiples of the residue
characteristic.
The number of n-th power classes, absolute-value form. For n with (n : K) ≠ 0, the
quotient Kˣ ⧸ (Kˣ)ⁿ has n · #μ_n(K) · ‖n‖_K⁻¹ elements, where μ_n(K) is the group of n-th
roots of unity in K and ‖·‖_K is the normalized absolute value of K. Since
‖n‖_K = q ^ (-v_K(n)), the factor ‖n‖_K⁻¹ is the factor q ^ v_K(n) of card_powerClasses,
and the equation holds in ℚ≥0 after casting the natural-number cardinalities.
For (n : K) ≠ 0, the subgroup (Kˣ)ⁿ of n-th powers has finite index in Kˣ.
The number of n-th power classes away from the residue characteristic. For n
invertible in 𝒪[K], the quotient Kˣ ⧸ (Kˣ)ⁿ has n · #μ_n(K) elements, where μ_n(K) is the
group of n-th roots of unity in K. This holds in either characteristic.
For n invertible in 𝒪[K], the subgroup (Kˣ)ⁿ of n-th powers has finite index in
Kˣ.
The square classes away from residue characteristic 2. If 2 is invertible in 𝒪[K],
then Kˣ ⧸ (Kˣ)² has 4 elements: μ_2(K) = {±1} has order 2.
The number of square classes of a nonarchimedean local field. If 2 is nonzero in K,
then Kˣ ⧸ (Kˣ)² has 4 · #𝓀[K] ^ v_K(2) elements.