mathlib documentation

field_theory.subfield

Subfields #

Let K be a field. This file defines the "bundled" subfield type subfield K, a type whose terms correspond to subfields of K. This is the preferred way to talk about subfields in mathlib. Unbundled subfields (s : set K and is_subfield s) are not in this file, and they will ultimately be deprecated.

We prove that subfields are a complete lattice, and that you can map (pushforward) and comap (pull back) them along ring homomorphisms.

We define the closure construction from set R to subfield R, sending a subset of R to the subfield it generates, and prove that it is a Galois insertion.

Main definitions #

Notation used here:

(K : Type u) [field K] (L : Type u) [field L] (f g : K →+* L) (A : subfield K) (B : subfield L) (s : set K)

subfield R : the type of subfields of a ring R.
instance : complete_lattice (subfield R) : the complete lattice structure on the subfields.
subfield.closure : subfield closure of a set, i.e., the smallest subfield that includes the set.
subfield.gi : closure : set M → subfield M and coercion coe : subfield M → set M form a galois_insertion.
comap f B : subfield K : the preimage of a subfield B along the ring homomorphism f
map f A : subfield L : the image of a subfield A along the ring homomorphism f.
prod A B : subfield (K × L) : the product of subfields
f.field_range : subfield B : the range of the ring homomorphism f.
eq_locus_field f g : subfield K : given ring homomorphisms f g : K →+* R, the subfield of K where f x = g x

Implementation notes #

A subfield is implemented as a subring which is is closed under ⁻¹.

Lattice inclusion (e.g. ≤ and ⊓) is used rather than set notation (⊆ and ∩), although ∈ is defined as membership of a subfield's underlying set.

Tags #

subfield, subfields

@[class]

structure subfield_class (S : Type u_1) (K : out_param (Type u_2)) [field K] [set_like S K] :

Type

to_subring_class : subring_class S K
to_inv_mem_class : inv_mem_class S K

subfield_class S K states S is a type of subsets s ⊆ K closed under field operations.

Instances

@[protected, instance]

def subfield_class.subfield_class.to_subgroup_class {K : Type u} [field K] (S : Type u_1) [set_like S K] [h : subfield_class S K] :

subgroup_class S K

A subfield contains 1, products and inverses.

Be assured that we're not actually proving that subfields are subgroups: subgroup_class is really an abbreviation of subgroup_with_or_without_zero_class.

Equations

subfield_class.subfield_class.to_subgroup_class S = {to_submonoid_class := subsemiring_class.to_submonoid_class subring_class.to_subsemiring_class, inv_mem := _}

@[protected, instance]

def subfield_class.to_field {K : Type u} [field K] (S : Type u_1) [set_like S K] [h : subfield_class S K] (s : S) :

A subfield inherits a field structure

Equations

subfield_class.to_field S s = function.injective.field coe _ _ _ _ _ _ _ _ _ _ _ _ _

@[protected, instance]

def subfield_class.to_linear_ordered_field (S : Type u_1) {K : Type u_2} [linear_ordered_field K] [set_like S K] [subfield_class S K] (s : S) :

linear_ordered_field ↥s

A subfield of a linear_ordered_field is a linear_ordered_field.

Equations

subfield_class.to_linear_ordered_field S s = function.injective.linear_ordered_field coe _ _ _ _ _ _ _ _ _ _ _ _ _

def subfield.to_subring {K : Type u} [field K] (self : subfield K) :

Reinterpret a subfield as a subring.

structure subfield (K : Type u) [field K] :

Type u

carrier : set K
mul_mem' : ∀ {a b : K}, a ∈ self.carrier → b ∈ self.carrier → a * b ∈ self.carrier
one_mem' : 1 ∈ self.carrier
add_mem' : ∀ {a b : K}, a ∈ self.carrier → b ∈ self.carrier → a + b ∈ self.carrier
zero_mem' : 0 ∈ self.carrier
neg_mem' : ∀ {x : K}, x ∈ self.carrier → -x ∈ self.carrier
inv_mem' : ∀ (x : K), x ∈ self.carrier → x⁻¹ ∈ self.carrier

subfield R is the type of subfields of R. A subfield of R is a subset s that is a multiplicative submonoid and an additive subgroup. Note in particular that it shares the same 0 and 1 as R.

def subfield.to_add_subgroup {K : Type u} [field K] (s : subfield K) :

The underlying add_subgroup of a subfield.

Equations

s.to_add_subgroup = {carrier := s.to_subring.to_add_subgroup.carrier, add_mem' := _, zero_mem' := _, neg_mem' := _}

def subfield.to_submonoid {K : Type u} [field K] (s : subfield K) :

The underlying submonoid of a subfield.

Equations

s.to_submonoid = {carrier := s.to_subring.to_submonoid.carrier, mul_mem' := _, one_mem' := _}

@[protected, instance]

def subfield.set_like {K : Type u} [field K] :

set_like (subfield K) K

Equations

subfield.set_like = {coe := subfield.carrier _inst_1, coe_injective' := _}

@[protected, instance]

def subfield.subfield_class {K : Type u} [field K] :

subfield_class (subfield K) K

Equations

subfield.subfield_class = subfield_class.mk

@[simp]

theorem subfield.mem_carrier {K : Type u} [field K] {s : subfield K} {x : K} :

x ∈ s.carrier ↔ x ∈ s

@[simp]

theorem subfield.mem_mk {K : Type u} [field K] {S : set K} {x : K} (h₁ : ∀ {a b : K}, a ∈ S → b ∈ S → a * b ∈ S) (h₂ : 1 ∈ S) (h₃ : ∀ {a b : K}, a ∈ S → b ∈ S → a + b ∈ S) (h₄ : 0 ∈ S) (h₅ : ∀ {x : K}, x ∈ S → -x ∈ S) (h₆ : ∀ (x : K), x ∈ S → x⁻¹ ∈ S) :

x ∈ {carrier := S, mul_mem' := h₁, one_mem' := h₂, add_mem' := h₃, zero_mem' := h₄, neg_mem' := h₅, inv_mem' := h₆} ↔ x ∈ S

@[simp]

theorem subfield.coe_set_mk {K : Type u} [field K] (S : set K) (h₁ : ∀ {a b : K}, a ∈ S → b ∈ S → a * b ∈ S) (h₂ : 1 ∈ S) (h₃ : ∀ {a b : K}, a ∈ S → b ∈ S → a + b ∈ S) (h₄ : 0 ∈ S) (h₅ : ∀ {x : K}, x ∈ S → -x ∈ S) (h₆ : ∀ (x : K), x ∈ S → x⁻¹ ∈ S) :

↑{carrier := S, mul_mem' := h₁, one_mem' := h₂, add_mem' := h₃, zero_mem' := h₄, neg_mem' := h₅, inv_mem' := h₆} = S

@[simp]

theorem subfield.mk_le_mk {K : Type u} [field K] {S S' : set K} (h₁ : ∀ {a b : K}, a ∈ S → b ∈ S → a * b ∈ S) (h₂ : 1 ∈ S) (h₃ : ∀ {a b : K}, a ∈ S → b ∈ S → a + b ∈ S) (h₄ : 0 ∈ S) (h₅ : ∀ {x : K}, x ∈ S → -x ∈ S) (h₆ : ∀ (x : K), x ∈ S → x⁻¹ ∈ S) (h₁' : ∀ {a b : K}, a ∈ S' → b ∈ S' → a * b ∈ S') (h₂' : 1 ∈ S') (h₃' : ∀ {a b : K}, a ∈ S' → b ∈ S' → a + b ∈ S') (h₄' : 0 ∈ S') (h₅' : ∀ {x : K}, x ∈ S' → -x ∈ S') (h₆' : ∀ (x : K), x ∈ S' → x⁻¹ ∈ S') :

{carrier := S, mul_mem' := h₁, one_mem' := h₂, add_mem' := h₃, zero_mem' := h₄, neg_mem' := h₅, inv_mem' := h₆} ≤ {carrier := S', mul_mem' := h₁', one_mem' := h₂', add_mem' := h₃', zero_mem' := h₄', neg_mem' := h₅', inv_mem' := h₆'} ↔ S ⊆ S'

@[ext]

theorem subfield.ext {K : Type u} [field K] {S T : subfield K} (h : ∀ (x : K), x ∈ S ↔ x ∈ T) :

S = T

Two subfields are equal if they have the same elements.

@[protected]

def subfield.copy {K : Type u} [field K] (S : subfield K) (s : set K) (hs : s = ↑S) :

Copy of a subfield with a new carrier equal to the old one. Useful to fix definitional equalities.

Equations

S.copy s hs = {carrier := s, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _, inv_mem' := _}

@[simp]

theorem subfield.coe_copy {K : Type u} [field K] (S : subfield K) (s : set K) (hs : s = ↑S) :

↑(S.copy s hs) = s

theorem subfield.copy_eq {K : Type u} [field K] (S : subfield K) (s : set K) (hs : s = ↑S) :

S.copy s hs = S

@[simp]

theorem subfield.coe_to_subring {K : Type u} [field K] (s : subfield K) :

↑(s.to_subring) = ↑s

@[simp]

theorem subfield.mem_to_subring {K : Type u} [field K] (s : subfield K) (x : K) :

x ∈ s.to_subring ↔ x ∈ s

def subring.to_subfield {K : Type u} [field K] (s : subring K) (hinv : ∀ (x : K), x ∈ s → x⁻¹ ∈ s) :

A subring containing inverses is a subfield.

Equations

s.to_subfield hinv = {carrier := s.carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _, inv_mem' := hinv}

@[protected]

theorem subfield.one_mem {K : Type u} [field K] (s : subfield K) :

1 ∈ s

A subfield contains the field's 1.

@[protected]

theorem subfield.zero_mem {K : Type u} [field K] (s : subfield K) :

0 ∈ s

A subfield contains the field's 0.

@[protected]

theorem subfield.mul_mem {K : Type u} [field K] (s : subfield K) {x y : K} :

x ∈ s → y ∈ s → x * y ∈ s

A subfield is closed under multiplication.

@[protected]

theorem subfield.add_mem {K : Type u} [field K] (s : subfield K) {x y : K} :

x ∈ s → y ∈ s → x + y ∈ s

A subfield is closed under addition.

@[protected]

theorem subfield.neg_mem {K : Type u} [field K] (s : subfield K) {x : K} :

x ∈ s → -x ∈ s

A subfield is closed under negation.

@[protected]

theorem subfield.sub_mem {K : Type u} [field K] (s : subfield K) {x y : K} :

x ∈ s → y ∈ s → x - y ∈ s

A subfield is closed under subtraction.

@[protected]

theorem subfield.inv_mem {K : Type u} [field K] (s : subfield K) {x : K} :

x ∈ s → x⁻¹ ∈ s

A subfield is closed under inverses.

@[protected]

theorem subfield.div_mem {K : Type u} [field K] (s : subfield K) {x y : K} :

x ∈ s → y ∈ s → x / y ∈ s

A subfield is closed under division.

@[protected]

theorem subfield.list_prod_mem {K : Type u} [field K] (s : subfield K) {l : list K} :

(∀ (x : K), x ∈ l → x ∈ s) → l.prod ∈ s

Product of a list of elements in a subfield is in the subfield.

@[protected]

theorem subfield.list_sum_mem {K : Type u} [field K] (s : subfield K) {l : list K} :

(∀ (x : K), x ∈ l → x ∈ s) → l.sum ∈ s

Sum of a list of elements in a subfield is in the subfield.

@[protected]

theorem subfield.multiset_prod_mem {K : Type u} [field K] (s : subfield K) (m : multiset K) :

(∀ (a : K), a ∈ m → a ∈ s) → m.prod ∈ s

Product of a multiset of elements in a subfield is in the subfield.

@[protected]

theorem subfield.multiset_sum_mem {K : Type u} [field K] (s : subfield K) (m : multiset K) :

(∀ (a : K), a ∈ m → a ∈ s) → m.sum ∈ s

Sum of a multiset of elements in a subfield is in the subfield.

@[protected]

theorem subfield.prod_mem {K : Type u} [field K] (s : subfield K) {ι : Type u_1} {t : finset ι} {f : ι → K} (h : ∀ (c : ι), c ∈ t → f c ∈ s) :

∏ (i : ι) in t, f i ∈ s

Product of elements of a subfield indexed by a finset is in the subfield.

@[protected]

theorem subfield.sum_mem {K : Type u} [field K] (s : subfield K) {ι : Type u_1} {t : finset ι} {f : ι → K} (h : ∀ (c : ι), c ∈ t → f c ∈ s) :

∑ (i : ι) in t, f i ∈ s

Sum of elements in a subfield indexed by a finset is in the subfield.

@[protected]

theorem subfield.pow_mem {K : Type u} [field K] (s : subfield K) {x : K} (hx : x ∈ s) (n : ℕ) :

x ^ n ∈ s

@[protected]

theorem subfield.zsmul_mem {K : Type u} [field K] (s : subfield K) {x : K} (hx : x ∈ s) (n : ℤ) :

n • x ∈ s

@[protected]

theorem subfield.coe_int_mem {K : Type u} [field K] (s : subfield K) (n : ℤ) :

theorem subfield.zpow_mem {K : Type u} [field K] (s : subfield K) {x : K} (hx : x ∈ s) (n : ℤ) :

x ^ n ∈ s

@[protected, instance]

def subfield.ring {K : Type u} [field K] (s : subfield K) :

Equations

s.ring = s.to_subring.to_ring

@[protected, instance]

def subfield.has_div {K : Type u} [field K] (s : subfield K) :

Equations

s.has_div = {div := λ (x y : ↥s), ⟨↑x / ↑y, _⟩}

@[protected, instance]

def subfield.has_inv {K : Type u} [field K] (s : subfield K) :

Equations

s.has_inv = {inv := λ (x : ↥s), ⟨(↑x)⁻¹, _⟩}

@[protected, instance]

def subfield.int.has_pow {K : Type u} [field K] (s : subfield K) :

has_pow ↥s ℤ

Equations

subfield.int.has_pow s = {pow := λ (x : ↥s) (z : ℤ), ⟨↑x ^ z, _⟩}

@[protected, instance]

def subfield.to_field {K : Type u} [field K] (s : subfield K) :

A subfield inherits a field structure

Equations

s.to_field = function.injective.field coe _ _ _ _ _ _ _ _ _ _ _ _ _

@[protected, instance]

def subfield.to_linear_ordered_field {K : Type u_1} [linear_ordered_field K] (s : subfield K) :

linear_ordered_field ↥s

A subfield of a linear_ordered_field is a linear_ordered_field.

Equations

s.to_linear_ordered_field = function.injective.linear_ordered_field coe _ _ _ _ _ _ _ _ _ _ _ _ _

@[simp, norm_cast]

theorem subfield.coe_add {K : Type u} [field K] (s : subfield K) (x y : ↥s) :

↑(x + y) = ↑x + ↑y

@[simp, norm_cast]

theorem subfield.coe_sub {K : Type u} [field K] (s : subfield K) (x y : ↥s) :

↑(x - y) = ↑x - ↑y

@[simp, norm_cast]

theorem subfield.coe_neg {K : Type u} [field K] (s : subfield K) (x : ↥s) :

@[simp, norm_cast]

theorem subfield.coe_mul {K : Type u} [field K] (s : subfield K) (x y : ↥s) :

↑x * y = (↑x) * ↑y

@[simp, norm_cast]

theorem subfield.coe_div {K : Type u} [field K] (s : subfield K) (x y : ↥s) :

↑(x / y) = ↑x / ↑y

@[simp, norm_cast]

theorem subfield.coe_inv {K : Type u} [field K] (s : subfield K) (x : ↥s) :

↑x⁻¹ = (↑x)⁻¹

@[simp, norm_cast]

theorem subfield.coe_zero {K : Type u} [field K] (s : subfield K) :

↑0 = 0

@[simp, norm_cast]

theorem subfield.coe_one {K : Type u} [field K] (s : subfield K) :

↑1 = 1

def subfield.subtype {K : Type u} [field K] (s : subfield K) :

The embedding from a subfield of the field K to K.

Equations

s.subtype = {to_fun := coe coe_to_lift, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

@[protected, instance]

def subfield.to_algebra {K : Type u} [field K] (s : subfield K) :

Equations

s.to_algebra = s.subtype.to_algebra

@[simp]

theorem subfield.coe_subtype {K : Type u} [field K] (s : subfield K) :

⇑(s.subtype) = coe

theorem subfield.to_subring.subtype_eq_subtype (F : Type u_1) [field F] (S : subfield F) :

S.to_subring.subtype = S.subtype

Partial order #

@[simp]

theorem subfield.mem_to_submonoid {K : Type u} [field K] {s : subfield K} {x : K} :

x ∈ s.to_submonoid ↔ x ∈ s

@[simp]

theorem subfield.coe_to_submonoid {K : Type u} [field K] (s : subfield K) :

↑(s.to_submonoid) = ↑s

@[simp]

theorem subfield.mem_to_add_subgroup {K : Type u} [field K] {s : subfield K} {x : K} :

x ∈ s.to_add_subgroup ↔ x ∈ s

@[simp]

theorem subfield.coe_to_add_subgroup {K : Type u} [field K] (s : subfield K) :

↑(s.to_add_subgroup) = ↑s

top #

@[protected, instance]

def subfield.has_top {K : Type u} [field K] :

has_top (subfield K)

The subfield of K containing all elements of K.

Equations

subfield.has_top = {top := {carrier := ⊤.carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _, inv_mem' := _}}

@[protected, instance]

def subfield.inhabited {K : Type u} [field K] :

inhabited (subfield K)

Equations

subfield.inhabited = {default := ⊤}

@[simp]

theorem subfield.mem_top {K : Type u} [field K] (x : K) :

@[simp]

theorem subfield.coe_top {K : Type u} [field K] :

↑⊤ = set.univ

comap #

def subfield.comap {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) (s : subfield L) :

The preimage of a subfield along a ring homomorphism is a subfield.

Equations

subfield.comap f s = {carrier := (subring.comap f s.to_subring).carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _, inv_mem' := _}

@[simp]

theorem subfield.coe_comap {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) (s : subfield L) :

↑(subfield.comap f s) = ⇑f ⁻¹' ↑s

@[simp]

theorem subfield.mem_comap {K : Type u} {L : Type v} [field K] [field L] {s : subfield L} {f : K →+* L} {x : K} :

x ∈ subfield.comap f s ↔ ⇑f x ∈ s

theorem subfield.comap_comap {K : Type u} {L : Type v} {M : Type w} [field K] [field L] [field M] (s : subfield M) (g : L →+* M) (f : K →+* L) :

subfield.comap f (subfield.comap g s) = subfield.comap (g.comp f) s

map #

def subfield.map {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) (s : subfield K) :

The image of a subfield along a ring homomorphism is a subfield.

Equations

subfield.map f s = {carrier := (subring.map f s.to_subring).carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _, inv_mem' := _}

@[simp]

theorem subfield.coe_map {K : Type u} {L : Type v} [field K] [field L] (s : subfield K) (f : K →+* L) :

↑(subfield.map f s) = ⇑f '' ↑s

@[simp]

theorem subfield.mem_map {K : Type u} {L : Type v} [field K] [field L] {f : K →+* L} {s : subfield K} {y : L} :

y ∈ subfield.map f s ↔ ∃ (x : K) (H : x ∈ s), ⇑f x = y

theorem subfield.map_map {K : Type u} {L : Type v} {M : Type w} [field K] [field L] [field M] (s : subfield K) (g : L →+* M) (f : K →+* L) :

subfield.map g (subfield.map f s) = subfield.map (g.comp f) s

theorem subfield.map_le_iff_le_comap {K : Type u} {L : Type v} [field K] [field L] {f : K →+* L} {s : subfield K} {t : subfield L} :

subfield.map f s ≤ t ↔ s ≤ subfield.comap f t

theorem subfield.gc_map_comap {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) :

galois_connection (subfield.map f) (subfield.comap f)

range #

def ring_hom.field_range {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) :

The range of a ring homomorphism, as a subfield of the target. See Note [range copy pattern].

Equations

f.field_range = (subfield.map f ⊤).copy (set.range ⇑f) _

@[simp]

theorem ring_hom.coe_field_range {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) :

↑(f.field_range) = set.range ⇑f

@[simp]

theorem ring_hom.mem_field_range {K : Type u} {L : Type v} [field K] [field L] {f : K →+* L} {y : L} :

y ∈ f.field_range ↔ ∃ (x : K), ⇑f x = y

theorem ring_hom.field_range_eq_map {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) :

f.field_range = subfield.map f ⊤

theorem ring_hom.map_field_range {K : Type u} {L : Type v} {M : Type w} [field K] [field L] [field M] (g : L →+* M) (f : K →+* L) :

subfield.map g f.field_range = (g.comp f).field_range

@[protected, instance]

def ring_hom.fintype_field_range {K : Type u} {L : Type v} [field K] [field L] [fintype K] [decidable_eq L] (f : K →+* L) :

fintype ↥(f.field_range)

The range of a morphism of fields is a fintype, if the domain is a fintype.

Note that this instance can cause a diamond with subtype.fintype if L is also a fintype.

Equations

f.fintype_field_range = set.fintype_range ⇑f

inf #

@[protected, instance]

def subfield.has_inf {K : Type u} [field K] :

has_inf (subfield K)

The inf of two subfields is their intersection.

Equations

subfield.has_inf = {inf := λ (s t : subfield K), {carrier := (s.to_subring ⊓ t.to_subring).carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _, inv_mem' := _}}

@[simp]

theorem subfield.coe_inf {K : Type u} [field K] (p p' : subfield K) :

↑(p ⊓ p') = ↑p ∩ ↑p'

@[simp]

theorem subfield.mem_inf {K : Type u} [field K] {p p' : subfield K} {x : K} :

x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

@[protected, instance]

def subfield.has_Inf {K : Type u} [field K] :

has_Inf (subfield K)

Equations

subfield.has_Inf = {Inf := λ (S : set (subfield K)), {carrier := (Inf (subfield.to_subring '' S)).carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _, inv_mem' := _}}

@[simp, norm_cast]

theorem subfield.coe_Inf {K : Type u} [field K] (S : set (subfield K)) :

↑(Inf S) = ⋂ (s : subfield K) (H : s ∈ S), ↑s

theorem subfield.mem_Inf {K : Type u} [field K] {S : set (subfield K)} {x : K} :

x ∈ Inf S ↔ ∀ (p : subfield K), p ∈ S → x ∈ p

@[simp]

theorem subfield.Inf_to_subring {K : Type u} [field K] (s : set (subfield K)) :

(Inf s).to_subring = ⨅ (t : subfield K) (H : t ∈ s), t.to_subring

theorem subfield.is_glb_Inf {K : Type u} [field K] (S : set (subfield K)) :

is_glb S (Inf S)

@[protected, instance]

def subfield.complete_lattice {K : Type u} [field K] :

complete_lattice (subfield K)

Subfields of a ring form a complete lattice.

Equations

subfield.complete_lattice = {sup := complete_lattice.sup (complete_lattice_of_Inf (subfield K) subfield.is_glb_Inf), le := complete_lattice.le (complete_lattice_of_Inf (subfield K) subfield.is_glb_Inf), lt := complete_lattice.lt (complete_lattice_of_Inf (subfield K) subfield.is_glb_Inf), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := has_inf.inf subfield.has_inf, inf_le_left := _, inf_le_right := _, le_inf := _, Sup := complete_lattice.Sup (complete_lattice_of_Inf (subfield K) subfield.is_glb_Inf), le_Sup := _, Sup_le := _, Inf := complete_lattice.Inf (complete_lattice_of_Inf (subfield K) subfield.is_glb_Inf), Inf_le := _, le_Inf := _, top := ⊤, bot := complete_lattice.bot (complete_lattice_of_Inf (subfield K) subfield.is_glb_Inf), le_top := _, bot_le := _}

subfield closure of a subset #

def subfield.closure {K : Type u} [field K] (s : set K) :

The subfield generated by a set.

Equations

subfield.closure s = {carrier := {_x : K | ∃ (x : K) (H : x ∈ subring.closure s) (y : K) (H : y ∈ subring.closure s), x / y = _x}, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _, inv_mem' := _}

theorem subfield.mem_closure_iff {K : Type u} [field K] {s : set K} {x : K} :

x ∈ subfield.closure s ↔ ∃ (y : K) (H : y ∈ subring.closure s) (z : K) (H : z ∈ subring.closure s), y / z = x

theorem subfield.subring_closure_le {K : Type u} [field K] (s : set K) :

subring.closure s ≤ (subfield.closure s).to_subring

@[simp]

theorem subfield.subset_closure {K : Type u} [field K] {s : set K} :

s ⊆ ↑(subfield.closure s)

The subfield generated by a set includes the set.

theorem subfield.not_mem_of_not_mem_closure {K : Type u} [field K] {s : set K} {P : K} (hP : P ∉ subfield.closure s) :

P ∉ s

theorem subfield.mem_closure {K : Type u} [field K] {x : K} {s : set K} :

x ∈ subfield.closure s ↔ ∀ (S : subfield K), s ⊆ ↑S → x ∈ S

@[simp]

theorem subfield.closure_le {K : Type u} [field K] {s : set K} {t : subfield K} :

subfield.closure s ≤ t ↔ s ⊆ ↑t

A subfield t includes closure s if and only if it includes s.

theorem subfield.closure_mono {K : Type u} [field K] ⦃s t : set K⦄ (h : s ⊆ t) :

subfield.closure s ≤ subfield.closure t

Subfield closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t.

theorem subfield.closure_eq_of_le {K : Type u} [field K] {s : set K} {t : subfield K} (h₁ : s ⊆ ↑t) (h₂ : t ≤ subfield.closure s) :

subfield.closure s = t

theorem subfield.closure_induction {K : Type u} [field K] {s : set K} {p : K → Prop} {x : K} (h : x ∈ subfield.closure s) (Hs : ∀ (x : K), x ∈ s → p x) (H1 : p 1) (Hadd : ∀ (x y : K), p x → p y → p (x + y)) (Hneg : ∀ (x : K), p x → p (-x)) (Hinv : ∀ (x : K), p x → p x⁻¹) (Hmul : ∀ (x y : K), p x → p y → p (x * y)) :

p x

An induction principle for closure membership. If p holds for 1, and all elements of s, and is preserved under addition, negation, and multiplication, then p holds for all elements of the closure of s.

@[protected]

def subfield.gi (K : Type u) [field K] :

galois_insertion subfield.closure coe

closure forms a Galois insertion with the coercion to set.

Equations

subfield.gi K = {choice := λ (s : set K) (_x : ↑(subfield.closure s) ≤ s), subfield.closure s, gc := _, le_l_u := _, choice_eq := _}

theorem subfield.closure_eq {K : Type u} [field K] (s : subfield K) :

subfield.closure ↑s = s

Closure of a subfield S equals S.

@[simp]

theorem subfield.closure_empty {K : Type u} [field K] :

subfield.closure ∅ = ⊥

@[simp]

theorem subfield.closure_univ {K : Type u} [field K] :

subfield.closure set.univ = ⊤

theorem subfield.closure_union {K : Type u} [field K] (s t : set K) :

subfield.closure (s ∪ t) = subfield.closure s ⊔ subfield.closure t

theorem subfield.closure_Union {K : Type u} [field K] {ι : Sort u_1} (s : ι → set K) :

subfield.closure (⋃ (i : ι), s i) = ⨆ (i : ι), subfield.closure (s i)

theorem subfield.closure_sUnion {K : Type u} [field K] (s : set (set K)) :

subfield.closure (⋃₀s) = ⨆ (t : set K) (H : t ∈ s), subfield.closure t

theorem subfield.map_sup {K : Type u} {L : Type v} [field K] [field L] (s t : subfield K) (f : K →+* L) :

subfield.map f (s ⊔ t) = subfield.map f s ⊔ subfield.map f t

theorem subfield.map_supr {K : Type u} {L : Type v} [field K] [field L] {ι : Sort u_1} (f : K →+* L) (s : ι → subfield K) :

subfield.map f (supr s) = ⨆ (i : ι), subfield.map f (s i)

theorem subfield.comap_inf {K : Type u} {L : Type v} [field K] [field L] (s t : subfield L) (f : K →+* L) :

subfield.comap f (s ⊓ t) = subfield.comap f s ⊓ subfield.comap f t

theorem subfield.comap_infi {K : Type u} {L : Type v} [field K] [field L] {ι : Sort u_1} (f : K →+* L) (s : ι → subfield L) :

subfield.comap f (infi s) = ⨅ (i : ι), subfield.comap f (s i)

@[simp]

theorem subfield.map_bot {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) :

subfield.map f ⊥ = ⊥

@[simp]

theorem subfield.comap_top {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) :

subfield.comap f ⊤ = ⊤

theorem subfield.mem_supr_of_directed {K : Type u} [field K] {ι : Sort u_1} [hι : nonempty ι] {S : ι → subfield K} (hS : directed has_le.le S) {x : K} :

(x ∈ ⨆ (i : ι), S i) ↔ ∃ (i : ι), x ∈ S i

The underlying set of a non-empty directed Sup of subfields is just a union of the subfields. Note that this fails without the directedness assumption (the union of two subfields is typically not a subfield)

theorem subfield.coe_supr_of_directed {K : Type u} [field K] {ι : Sort u_1} [hι : nonempty ι] {S : ι → subfield K} (hS : directed has_le.le S) :

(↑⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

theorem subfield.mem_Sup_of_directed_on {K : Type u} [field K] {S : set (subfield K)} (Sne : S.nonempty) (hS : directed_on has_le.le S) {x : K} :

x ∈ Sup S ↔ ∃ (s : subfield K) (H : s ∈ S), x ∈ s

theorem subfield.coe_Sup_of_directed_on {K : Type u} [field K] {S : set (subfield K)} (Sne : S.nonempty) (hS : directed_on has_le.le S) :

↑(Sup S) = ⋃ (s : subfield K) (H : s ∈ S), ↑s

def ring_hom.cod_restrict_field {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) (s : subfield L) (h : ∀ (x : K), ⇑f x ∈ s) :

Restrict the codomain of a ring homomorphism to a subfield that includes the range.

Equations

f.cod_restrict_field s h = {to_fun := λ (x : K), ⟨⇑f x, _⟩, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

def ring_hom.restrict_field {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) (s : subfield K) :

Restriction of a ring homomorphism to a subfield of the domain.

Equations

f.restrict_field s = f.comp s.subtype

@[simp]

theorem ring_hom.restrict_field_apply {K : Type u} {L : Type v} [field K] [field L] {s : subfield K} (f : K →+* L) (x : ↥s) :

⇑(f.restrict_field s) x = ⇑f ↑x

def ring_hom.range_restrict_field {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) :

K →+* ↥(f.field_range)

Restriction of a ring homomorphism to its range interpreted as a subfield.

Equations

f.range_restrict_field = f.srange_restrict

@[simp]

theorem ring_hom.coe_range_restrict_field {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) (x : K) :

↑(⇑(f.range_restrict_field) x) = ⇑f x

def ring_hom.eq_locus_field {K : Type u} {L : Type v} [field K] [field L] (f g : K →+* L) :

The subfield of elements x : R such that f x = g x, i.e., the equalizer of f and g as a subfield of R

Equations

f.eq_locus_field g = {carrier := {x : K | ⇑f x = ⇑g x}, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _, inv_mem' := _}

theorem ring_hom.eq_on_field_closure {K : Type u} {L : Type v} [field K] [field L] {f g : K →+* L} {s : set K} (h : set.eq_on ⇑f ⇑g s) :

set.eq_on ⇑f ⇑g ↑(subfield.closure s)

If two ring homomorphisms are equal on a set, then they are equal on its subfield closure.

theorem ring_hom.eq_of_eq_on_subfield_top {K : Type u} {L : Type v} [field K] [field L] {f g : K →+* L} (h : set.eq_on ⇑f ⇑g ↑⊤) :

f = g

theorem ring_hom.eq_of_eq_on_of_field_closure_eq_top {K : Type u} {L : Type v} [field K] [field L] {s : set K} (hs : subfield.closure s = ⊤) {f g : K →+* L} (h : set.eq_on ⇑f ⇑g s) :

f = g

theorem ring_hom.field_closure_preimage_le {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) (s : set L) :

subfield.closure (⇑f ⁻¹' s) ≤ subfield.comap f (subfield.closure s)

theorem ring_hom.map_field_closure {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) (s : set K) :

subfield.map f (subfield.closure s) = subfield.closure (⇑f '' s)

The image under a ring homomorphism of the subfield generated by a set equals the subfield generated by the image of the set.

def subfield.inclusion {K : Type u} [field K] {S T : subfield K} (h : S ≤ T) :

↥S →+* ↥T

The ring homomorphism associated to an inclusion of subfields.

Equations

subfield.inclusion h = S.subtype.cod_restrict_field T _

@[simp]

theorem subfield.field_range_subtype {K : Type u} [field K] (s : subfield K) :

s.subtype.field_range = s

def ring_equiv.subfield_congr {K : Type u} [field K] {s t : subfield K} (h : s = t) :

↥s ≃+* ↥t

Makes the identity isomorphism from a proof two subfields of a multiplicative monoid are equal.

Equations

ring_equiv.subfield_congr h = {to_fun := (equiv.set_congr _).to_fun, inv_fun := (equiv.set_congr _).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

theorem subfield.closure_preimage_le {K : Type u} {L : Type v} [field K] [field L] (f : K →+* L) (s : set L) :

subfield.closure (⇑f ⁻¹' s) ≤ subfield.comap f (subfield.closure s)