This repository has been archived by the owner on Jul 24, 2024. It is now read-only.
-
Notifications
You must be signed in to change notification settings - Fork 298
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
feat(field_theory): intermediate fields
Define `intermediate_field K L` as a structure extending `subalgebra K L` and `subfield L`. Requires re-bundling `subalgebra`, which will be another PR.
- Loading branch information
1 parent
ae118cd
commit 8baee2b
Showing
3 changed files
with
258 additions
and
1 deletion.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,254 @@ | ||
/- | ||
Copyright (c) 2020 Anne Baanen. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Anne Baanen | ||
-/ | ||
|
||
import field_theory.subfield | ||
import ring_theory.algebra_tower | ||
|
||
/-! | ||
# Intermediate fields | ||
Let `L / K` be a field extension, given as an instance `algebra K L`. | ||
This file defines the type of fields in between `K` and `L`, `intermediate_field K L`. | ||
An `intermediate_field K L` is a subfield of `L` which contains (the image of) `K`, | ||
i.e. it is a `subfield L` and a `subalgebra K L`. | ||
## Main definitions | ||
* `intermediate_field K L` : the type of intermediate fields between `K` and `L`. | ||
* `subalgebra.to_intermediate_field`: turns a subalgebra closed under `⁻¹` | ||
into an intermediate field | ||
* `subfield.to_intermediate_field`: turns a subfield containing the image of `K` | ||
into an intermediate field | ||
* `intermediate_field.map`: map an intermediate field along an `alg_hom` | ||
## Implementation notes | ||
Intermediate fields are defined with a structure extending `subfield` and `subalgebra`. | ||
A `subalgebra` is closed under all operations except `⁻¹`, | ||
## TODO | ||
* `field.adjoin` currently returns a `subalgebra`, this should become an | ||
`intermediate_field`. The lattice structure on `intermediate_field` will | ||
follow from the adjunction given by `field.adjoin`. | ||
## Tags | ||
intermediate field, field extension | ||
-/ | ||
|
||
open_locale big_operators | ||
|
||
variables (K L : Type*) [field K] [field L] [algebra K L] | ||
|
||
section | ||
set_option old_structure_cmd true | ||
|
||
/-- `S : intermediate_field K L` is a subset of `L` such that there is a field | ||
tower `L / S / K`. -/ | ||
structure intermediate_field extends subalgebra K L, subfield L | ||
|
||
/-- Reinterpret an `intermediate_field` as a `subalgebra`. -/ | ||
add_decl_doc intermediate_field.to_subalgebra | ||
|
||
/-- Reinterpret an `intermediate_field` as a `subfield`. -/ | ||
add_decl_doc intermediate_field.to_subfield | ||
|
||
end | ||
|
||
variables {K L} (S : intermediate_field K L) | ||
|
||
namespace intermediate_field | ||
|
||
instance : has_coe (intermediate_field K L) (set L) := | ||
⟨intermediate_field.carrier⟩ | ||
|
||
@[simp] lemma coe_to_subalgebra : (S.to_subalgebra : set L) = S := rfl | ||
|
||
@[simp] lemma coe_to_subfield : (S.to_subfield : set L) = S := rfl | ||
|
||
instance : has_coe_to_sort (intermediate_field K L) := ⟨Type*, λ S, S.carrier⟩ | ||
|
||
instance : has_mem L (intermediate_field K L) := ⟨λ m S, m ∈ (S : set L)⟩ | ||
|
||
@[simp] lemma mem_mk (s : set L) (hK : ∀ x, algebra_map K L x ∈ s) | ||
(ho hm hz ha hn hi) (x : L) : | ||
x ∈ intermediate_field.mk s ho hm hz ha hK hn hi ↔ x ∈ s := iff.rfl | ||
|
||
@[simp] lemma mem_coe (x : L) : x ∈ (S : set L) ↔ x ∈ S := iff.rfl | ||
|
||
@[simp] lemma mem_to_subalgebra (s : intermediate_field K L) (x : L) : | ||
x ∈ s.to_subalgebra ↔ x ∈ s := iff.rfl | ||
|
||
@[simp] lemma mem_to_subfield (s : intermediate_field K L) (x : L) : | ||
x ∈ s.to_subfield ↔ x ∈ s := iff.rfl | ||
|
||
/-- Two intermediate fields are equal if the underlying subsets are equal. -/ | ||
theorem ext' ⦃s t : intermediate_field K L⦄ (h : (s : set L) = t) : s = t := | ||
by { cases s, cases t, congr' } | ||
|
||
/-- Two intermediate fields are equal if and only if the underlying subsets are equal. -/ | ||
protected theorem ext'_iff {s t : intermediate_field K L} : s = t ↔ (s : set L) = t := | ||
⟨λ h, h ▸ rfl, λ h, ext' h⟩ | ||
|
||
/-- Two intermediate fields are equal if they have the same elements. -/ | ||
@[ext] theorem ext {S T : intermediate_field K L} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T := | ||
ext' $ set.ext h | ||
|
||
/-- An intermediate field contains the image of the smaller field. -/ | ||
theorem algebra_map_mem (x : K) : algebra_map K L x ∈ S := | ||
S.algebra_map_mem' x | ||
|
||
/-- An intermediate field contains the ring's 1. -/ | ||
theorem one_mem : (1 : L) ∈ S := S.one_mem' | ||
|
||
/-- An intermediate field contains the ring's 0. -/ | ||
theorem zero_mem : (0 : L) ∈ S := S.zero_mem' | ||
|
||
/-- An intermediate field is closed under multiplication. -/ | ||
theorem mul_mem : ∀ {x y : L}, x ∈ S → y ∈ S → x * y ∈ S := S.mul_mem' | ||
|
||
/-- An intermediate field is closed under scalar multiplication. -/ | ||
theorem smul_mem {y : L} : y ∈ S → ∀ {x : K}, x • y ∈ S := S.to_subalgebra.smul_mem | ||
|
||
/-- An intermediate field is closed under addition. -/ | ||
theorem add_mem : ∀ {x y : L}, x ∈ S → y ∈ S → x + y ∈ S := S.add_mem' | ||
|
||
/-- An intermediate field is closed under subtraction -/ | ||
theorem sub_mem {x y : L} (hx : x ∈ S) (hy : y ∈ S) : x - y ∈ S := | ||
S.to_subfield.sub_mem hx hy | ||
|
||
/-- An intermediate field is closed under negation. -/ | ||
theorem neg_mem : ∀ {x : L}, x ∈ S → -x ∈ S := S.neg_mem' | ||
|
||
/-- An intermediate field is closed under inverses. -/ | ||
theorem inv_mem : ∀ {x : L}, x ∈ S → x⁻¹ ∈ S := S.inv_mem' | ||
|
||
/-- An intermediate field is closed under division. -/ | ||
theorem div_mem {x y : L} (hx : x ∈ S) (hy : y ∈ S) : x / y ∈ S := | ||
S.to_subfield.div_mem hx hy | ||
|
||
/-- Product of a list of elements in an intermediate_field is in the intermediate_field. -/ | ||
lemma list_prod_mem {l : list L} : (∀ x ∈ l, x ∈ S) → l.prod ∈ S := | ||
S.to_subfield.list_prod_mem | ||
|
||
/-- Sum of a list of elements in an intermediate field is in the intermediate_field. -/ | ||
lemma list_sum_mem {l : list L} : (∀ x ∈ l, x ∈ S) → l.sum ∈ S := | ||
S.to_subfield.list_sum_mem | ||
|
||
/-- Product of a multiset of elements in an intermediate field is in the intermediate_field. -/ | ||
lemma multiset_prod_mem (m : multiset L) : | ||
(∀ a ∈ m, a ∈ S) → m.prod ∈ S := | ||
S.to_subfield.multiset_prod_mem m | ||
|
||
/-- Sum of a multiset of elements in a `intermediate_field` is in the `intermediate_field`. -/ | ||
lemma multiset_sum_mem (m : multiset L) : | ||
(∀ a ∈ m, a ∈ S) → m.sum ∈ S := | ||
S.to_subfield.multiset_sum_mem m | ||
|
||
/-- Product of elements of an intermediate field indexed by a `finset` is in the intermediate_field. -/ | ||
lemma prod_mem {ι : Type*} {t : finset ι} {f : ι → L} (h : ∀ c ∈ t, f c ∈ S) : | ||
∏ i in t, f i ∈ S := | ||
S.to_subfield.prod_mem h | ||
|
||
/-- Sum of elements in a `intermediate_field` indexed by a `finset` is in the `intermediate_field`. -/ | ||
lemma sum_mem {ι : Type*} {t : finset ι} {f : ι → L} (h : ∀ c ∈ t, f c ∈ S) : | ||
∑ i in t, f i ∈ S := | ||
S.to_subfield.sum_mem h | ||
|
||
lemma pow_mem {x : L} (hx : x ∈ S) (n : ℕ) : x^n ∈ S := S.to_subfield.pow_mem hx n | ||
|
||
lemma gsmul_mem {x : L} (hx : x ∈ S) (n : ℤ) : | ||
n •ℤ x ∈ S := S.to_subfield.gsmul_mem hx n | ||
|
||
lemma coe_int_mem (n : ℤ) : (n : L) ∈ S := | ||
by simp only [← gsmul_one, gsmul_mem, one_mem] | ||
|
||
instance : inhabited (intermediate_field K L) := | ||
⟨{ neg_mem' := λ x hx, (⊤ : subalgebra K L).neg_mem hx, | ||
inv_mem' := λ x hx, algebra.mem_top, | ||
..(⊤ : subalgebra K L) }⟩ | ||
|
||
end intermediate_field | ||
|
||
/-- Turn a subalgebra closed under inverses into an intermediate field -/ | ||
def subalgebra.to_intermediate_field (S : subalgebra K L) (inv_mem : ∀ x ∈ S, x⁻¹ ∈ S) : | ||
intermediate_field K L := | ||
{ neg_mem' := λ x, S.neg_mem, | ||
inv_mem' := inv_mem, | ||
.. S } | ||
|
||
/-- Turn a subfield of `L` containing the image of `K` into an intermediate field -/ | ||
def subfield.to_intermediate_field (S : subfield L) | ||
(algebra_map_mem : ∀ x, algebra_map K L x ∈ S) : | ||
intermediate_field K L := | ||
{ algebra_map_mem' := algebra_map_mem, | ||
.. S } | ||
|
||
namespace intermediate_field | ||
|
||
/-- An intermediate field inherits a field structure -/ | ||
instance to_field : field S := | ||
S.to_subfield.to_field | ||
|
||
@[simp, norm_cast] lemma coe_add (x y : S) : (↑(x + y) : L) = ↑x + ↑y := rfl | ||
@[simp, norm_cast] lemma coe_neg (x : S) : (↑(-x) : L) = -↑x := rfl | ||
@[simp, norm_cast] lemma coe_mul (x y : S) : (↑(x * y) : L) = ↑x * ↑y := rfl | ||
@[simp, norm_cast] lemma coe_inv (x : S) : (↑(x⁻¹) : L) = (↑x)⁻¹ := rfl | ||
@[simp, norm_cast] lemma coe_zero : ((0 : S) : L) = 0 := rfl | ||
@[simp, norm_cast] lemma coe_one : ((1 : S) : L) = 1 := rfl | ||
|
||
instance algebra : algebra K S := | ||
S.to_subalgebra.algebra | ||
|
||
instance to_algebra : algebra S L := | ||
S.to_subalgebra.to_algebra | ||
|
||
instance : is_scalar_tower K S L := | ||
is_scalar_tower.subalgebra _ _ S.to_subalgebra | ||
|
||
variables {L' : Type*} [field L'] [algebra K L'] | ||
|
||
/-- If `f : L →+* L'` fixes `K`, `S.map f` is the intermediate field between `L'` and `K` | ||
such that `x ∈ S ↔ f x ∈ S.map f`. -/ | ||
def map (f : L →ₐ[K] L') : intermediate_field K L' := | ||
{ inv_mem' := by { rintros _ ⟨x, hx, rfl⟩, exact ⟨x⁻¹, S.inv_mem hx, f.map_inv x⟩ }, | ||
neg_mem' := λ x hx, (S.to_subalgebra.map f).neg_mem hx, | ||
.. S.to_subalgebra.map f} | ||
|
||
/-- The embedding from an intermediate field of `L / K` to `L`. -/ | ||
def val : S →ₐ[K] L := | ||
S.to_subalgebra.val | ||
|
||
@[simp] theorem coe_val : ⇑S.val = coe := rfl | ||
|
||
@[simp] lemma val_mk {x : L} (hx : x ∈ S) : S.val ⟨x, hx⟩ = x := rfl | ||
|
||
variables {S} | ||
|
||
lemma to_subalgebra_injective {S S' : intermediate_field K L} | ||
(h : S.to_subalgebra = S'.to_subalgebra) : S = S' := | ||
by { ext, rw [← mem_to_subalgebra, ← mem_to_subalgebra, h] } | ||
|
||
instance : partial_order (intermediate_field K L) := | ||
{ le := λ S T, (S : set L) ⊆ T, | ||
le_refl := λ S, set.subset.refl S, | ||
le_trans := λ _ _ _, set.subset.trans, | ||
le_antisymm := λ S T hst hts, ext $ λ x, ⟨@hst x, @hts x⟩ } | ||
|
||
variables (S) | ||
|
||
lemma set_range_subset : set.range (algebra_map K L) ⊆ S := | ||
S.to_subalgebra.range_subset | ||
|
||
lemma field_range_le : (algebra_map K L).field_range ≤ S.to_subfield := | ||
λ x hx, S.to_subalgebra.range_subset (by rwa [set.mem_range, ← ring_hom.mem_field_range]) | ||
|
||
variables {S} | ||
|
||
end intermediate_field |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters