mathlibdocumentation

field_theory.krull_topology

Krull topology #

We define the Krull topology on L ≃ₐ[K] L for an arbitrary field extension L/K. In order to do this, we first define a group_filter_basis on L ≃ₐ[K] L, whose sets are E.fixing_subgroup for all intermediate fields E with E/K finite dimensional.

Main Definitions #

• finite_exts K L. Given a field extension L/K, this is the set of intermediate fields that are finite-dimensional over K.

• fixed_by_finite K L. Given a field extension L/K, fixed_by_finite K L is the set of subsets Gal(L/E) of Gal(L/K), where E/K is finite

• gal_basis K L. Given a field extension L/K, this is the filter basis on L ≃ₐ[K] L whose sets are Gal(L/E) for intermediate fields E with E/K finite.

• gal_group_basis K L. This is the same as gal_basis K L, but with the added structure that it is a group filter basis on L ≃ₐ[K] L, rather than just a filter basis.

• krull_topology K L. Given a field extension L/K, this is the topology on L ≃ₐ[K] L, induced by the group filter basis gal_group_basis K L.

Main Results #

• krull_topology_t2 K L. For an integral field extension L/K, the topology krull_topology K L is Hausdorff.

• krull_topology_totally_disconnected K L. For an integral field extension L/K, the topology krull_topology K L is totally disconnected.

Notations #

• In docstrings, we will write Gal(L/E) to denote the fixing subgroup of an intermediate field E. That is, Gal(L/E) is the subgroup of L ≃ₐ[K] L consisting of automorphisms that fix every element of E. In particular, we distinguish between L ≃ₐ[E] L and Gal(L/E), since the former is defined to be a subgroup of L ≃ₐ[K] L, while the latter is a group in its own right.

Implementation Notes #

• krull_topology K L is defined as an instance for type class inference.
theorem intermediate_field.map_mono {K : Type u_1} {L : Type u_2} {M : Type u_3} [field K] [field L] [field M] [ L] [ M] {E1 E2 : L} (e : L ≃ₐ[K] M) (h12 : E1 E2) :

Mapping intermediate fields along algebra equivalences preserves the partial order

theorem intermediate_field.map_id {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] (E : L) :
E.map L) = E

Mapping intermediate fields along the identity does not change them

@[protected, instance]
def im_finite_dimensional {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] {E : L} (σ : L ≃ₐ[K] L) [ E] :

Mapping a finite dimensional intermediate field along an algebra equivalence gives a finite-dimensional intermediate field.

def finite_exts (K : Type u_1) [field K] (L : Type u_2) [field L] [ L] :
set L)

Given a field extension L/K, finite_exts K L is the set of intermediate field extensions L/E/K such that E/K is finite

Equations
• L = {E : |
def fixed_by_finite (K : Type u_1) (L : Type u_2) [field K] [field L] [ L] :

Given a field extension L/K, fixed_by_finite K L is the set of subsets Gal(L/E) of L ≃ₐ[K] L, where E/K is finite

Equations
theorem intermediate_field.finite_dimensional_bot (K : Type u_1) (L : Type u_2) [field K] [field L] [ L] :

For an field extension L/K, the intermediate field K is finite-dimensional over K

theorem intermediate_field.fixing_subgroup.bot {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] :

This lemma says that Gal(L/K) = L ≃ₐ[K] L

theorem top_fixed_by_finite {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] :

If L/K is a field extension, then we have Gal(L/K) ∈ fixed_by_finite K L

theorem finite_dimensional_sup {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] (E1 E2 : L) (h1 : E1) (h2 : E2) :
(E1 E2)

If E1 and E2 are finite-dimensional intermediate fields, then so is their compositum. This rephrases a result already in mathlib so that it is compatible with our type classes

theorem intermediate_field.mem_fixing_subgroup_iff {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] (E : L) (σ : L ≃ₐ[K] L) :
∀ (x : L), x Eσ x = x

An element of L ≃ₐ[K] L is in Gal(L/E) if and only if it fixes every element of E

theorem intermediate_field.fixing_subgroup.antimono {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] {E1 E2 : L} (h12 : E1 E2) :

The map E ↦ Gal(L/E) is inclusion-reversing

def gal_basis (K : Type u_1) (L : Type u_2) [field K] [field L] [ L] :

Given a field extension L/K, gal_basis K L is the filter basis on L ≃ₐ[K] L whose sets are Gal(L/E) for intermediate fields E with E/K finite dimensional

Equations
theorem mem_gal_basis_iff (K : Type u_1) (L : Type u_2) [field K] [field L] [ L] (U : set (L ≃ₐ[K] L)) :
U L

A subset of L ≃ₐ[K] L is a member of gal_basis K L if and only if it is the underlying set of Gal(L/E) for some finite subextension E/K

def gal_group_basis (K : Type u_1) (L : Type u_2) [field K] [field L] [ L] :

For a field extension L/K, gal_group_basis K L is the group filter basis on L ≃ₐ[K] L whose sets are Gal(L/E) for finite subextensions E/K

Equations
@[protected, instance]
def krull_topology (K : Type u_1) (L : Type u_2) [field K] [field L] [ L] :

For a field extension L/K, krull_topology K L is the topological space structure on L ≃ₐ[K] L induced by the group filter basis gal_group_basis K L

Equations
@[protected, instance]
def alg_equiv.topological_group (K : Type u_1) (L : Type u_2) [field K] [field L] [ L] :

For a field extension L/K, the Krull topology on L ≃ₐ[K] L makes it a topological group.

theorem intermediate_field.fixing_subgroup_is_open {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] (E : L) [ E] :

Let L/E/K be a tower of fields with E/K finite. Then Gal(L/E) is an open subgroup of L ≃ₐ[K] L.

theorem intermediate_field.fixing_subgroup_is_closed {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] (E : L) [ E] :

Given a tower of fields L/E/K, with E/K finite, the subgroup Gal(L/E) ≤ L ≃ₐ[K] L is closed.

theorem krull_topology_t2 {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] (h_int : L) :

If L/K is an algebraic extension, then the Krull topology on L ≃ₐ[K] L is Hausdorff.

theorem krull_topology_totally_disconnected {K : Type u_1} {L : Type u_2} [field K] [field L] [ L] (h_int : L) :

If L/K is an algebraic field extension, then the Krull topology on L ≃ₐ[K] L is totally disconnected.