The structure of the multiplicative group of a local field #
For a nonarchimedean local field K with residue field of cardinality q, this file proves the
two splittings of the multiplicative group Kˣ:
- a choice of uniformizer
ϖgives an isomorphism of topological groupsKˣ ≃ₜ* ℤ × U(K,0),x ↦ (v_K(x), x ϖ^{-v_K(x)}), whereU(K,0) = 𝒪[K]ˣis the depth-zero step of the unit filtration and the first factor is the normalized valuation; - with no choice at all, the Teichmüller lift gives an isomorphism of topological groups
U(K,0) ≃ₜ* μ_{q-1}(K) × U(K,1), whose first component is the Teichmüller representative of the residue class.
Together they describe Kˣ as ℤ × μ_{q-1}(K) × U(K,1), which reduces questions about Kˣ,
such as the count of its power classes, to the group of principal units U(K,1).
Main definitions #
TauCeti.unitsEquivIntProd: the splittingKˣ ≃ₜ* ℤ × U(K,0)attached to a uniformizer.TauCeti.unitFiltrationZeroEquivProd: the splittingU(K,0) ≃ₜ* μ_{q-1}(K) × U(K,1).
Main results #
TauCeti.ker_normalizedValuationandTauCeti.continuous_normalizedValuation: the normalized valuation is a continuous homomorphism with kernelU(K,0).TauCeti.normalizedValuation_comp_zpowersHom: a uniformizer splits the normalized valuation.TauCeti.existsUnique_eq_zpow_mul: everyx : Kˣis uniquelyϖ ^ n * uwithu ∈ U(K,0).TauCeti.coe_unitsEquivIntProd_apply_snd_eq_mul: how the splitting changes with the uniformizer.TauCeti.rootsOfUnityAlgebraMulEquivUnitsResidueField_unitFiltrationZeroEquivProd_apply_fst: the root-of-unity component ofu ∈ U(K,0)has the same residue class asu.
Implementation notes #
A uniformizer is taken to be any ϖ : Kˣ with normalizedValuation K ϖ = ofAdd 1; an
irreducible element of 𝒪[K] provides one by TauCeti.normalizedValuation_irreducible. All the
groups involved are subgroups of Kˣ with the subspace topology, and ℤ is written
multiplicatively as Multiplicative ℤ, with its discrete topology.
References #
- J.-P. Serre, Corps Locaux, Chapter II, §§4–5.
- J. Neukirch, Algebraic Number Theory, Chapter II, §5, Proposition 5.3.
The kernel of the normalized valuation is the depth-zero step U(K,0) = 𝒪[K]ˣ of the unit
filtration.
The normalized valuation is continuous for the discrete topology on ℤ: it is constant on the
cosets of the open subgroup U(K,0).
The normalized valuation vanishes on U(K,0).
A uniformizer splits the normalized valuation: v_K(ϖ ^ n) = n.
The unit part x ϖ^{-v_K(x)} of x : Kˣ lies in U(K,0).
The structure of Kˣ attached to a uniformizer. For ϖ : Kˣ of normalized valuation
1, the map x ↦ (v_K(x), x ϖ^{-v_K(x)}) is an isomorphism of topological groups
Kˣ ≃ₜ* ℤ × U(K,0), with inverse (n, u) ↦ ϖ ^ n * u.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The ℤ-component of the uniformizer splitting is the normalized valuation.
The U(K,0)-component of the uniformizer splitting is x ϖ^{-v_K(x)}.
The inverse of the uniformizer splitting is (n, u) ↦ ϖ ^ n * u.
Uniqueness of the decomposition. Every x : Kˣ is uniquely ϖ ^ n * u with n : ℤ and
u ∈ U(K,0).
Changing the uniformizer. For two uniformizers ϖ and ϖ', the unit components of the
two splittings differ by the power (ϖ ϖ'⁻¹) ^ v_K(x) of the unit ϖ ϖ'⁻¹ ∈ U(K,0).
The ratio of two uniformizers lies in U(K,0).
Roots of unity have valuation one: μ_n(K) ≤ U(K,0) for n ≠ 0.
The Teichmüller splitting of U(K,0) = 𝒪[K]ˣ. The map sending u to its Teichmüller
representative ζ ∈ μ_{q-1}(K) together with the principal unit u ζ⁻¹ ∈ U(K,1) is an
isomorphism of topological groups U(K,0) ≃ₜ* μ_{q-1}(K) × U(K,1), with inverse
(ζ, v) ↦ ζ v.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The root-of-unity component of u ∈ U(K,0) has the same residue class as u.
The principal-unit component of u ∈ U(K,0) is u ζ⁻¹, for ζ the root-of-unity
component.
The inverse of the Teichmüller splitting is multiplication, (ζ, v) ↦ ζ v.