# Valued fields and their completions #

In this file we study the topology of a field K endowed with a valuation (in our application to adic spaces, K will be the valuation field associated to some valuation on a ring, defined in valuation.basic).

We already know from valuation.topology that one can build a topology on K which makes it a topological ring.

The first goal is to show K is a topological field, ie inversion is continuous at every non-zero element.

The next goal is to prove K is a completable topological field. This gives us a completion hat K which is a topological field. We also prove that K is automatically separated, so the map from K to hat K is injective.

Then we extend the valuation given on K to a valuation on hat K.

theorem Valuation.inversion_estimate {K : Type u_1} [] {Γ₀ : Type u_2} (v : Valuation K Γ₀) {x : K} {y : K} {γ : Γ₀ˣ} (y_ne : y 0) (h : v (x - y) < min (γ * (v y * v y)) (v y)) :
v (x⁻¹ - y⁻¹) < γ
@[instance 100]
instance Valued.topologicalDivisionRing {K : Type u_1} [] {Γ₀ : Type u_2} [Valued K Γ₀] :

The topology coming from a valuation on a division ring makes it a topological division ring [BouAC, VI.5.1 middle of Proposition 1]

Equations
• =
@[instance 100]
instance ValuedRing.separated {K : Type u_1} [] {Γ₀ : Type u_2} [Valued K Γ₀] :

A valued division ring is separated.

Equations
• =
theorem Valued.continuous_valuation {K : Type u_1} [] {Γ₀ : Type u_2} [Valued K Γ₀] :
Continuous Valued.v
@[instance 100]
instance Valued.completable {K : Type u_1} [] {Γ₀ : Type u_2} [hv : Valued K Γ₀] :

A valued field is completable.

Equations
• =
noncomputable def Valued.extension {K : Type u_1} [] {Γ₀ : Type u_2} [hv : Valued K Γ₀] :

The extension of the valuation of a valued field to the completion of the field.

Equations
• Valued.extension = .extend Valued.v
Instances For
theorem Valued.continuous_extension {K : Type u_1} [] {Γ₀ : Type u_2} [hv : Valued K Γ₀] :
Continuous Valued.extension
@[simp]
theorem Valued.extension_extends {K : Type u_1} [] {Γ₀ : Type u_2} [hv : Valued K Γ₀] (x : K) :
Valued.extension (K x) = Valued.v x
noncomputable def Valued.extensionValuation {K : Type u_1} [] {Γ₀ : Type u_2} [hv : Valued K Γ₀] :

the extension of a valuation on a division ring to its completion.

Equations
• Valued.extensionValuation = { toFun := Valued.extension, map_zero' := , map_one' := , map_mul' := , map_add_le_max' := }
Instances For
theorem Valued.closure_coe_completion_v_lt {K : Type u_1} [] {Γ₀ : Type u_2} [hv : Valued K Γ₀] {γ : Γ₀ˣ} :
closure (K '' {x : K | Valued.v x < γ}) = {x : | Valued.extensionValuation x < γ}
noncomputable instance Valued.valuedCompletion {K : Type u_1} [] {Γ₀ : Type u_2} [hv : Valued K Γ₀] :
Equations
• Valued.valuedCompletion = Valued.mk Valued.extensionValuation
@[simp]
theorem Valued.valuedCompletion_apply {K : Type u_1} [] {Γ₀ : Type u_2} [hv : Valued K Γ₀] (x : K) :
Valued.v (K x) = Valued.v x