Subfields

Let K be a division ring, for example a field. This file concerns the "bundled" subfield type Subfield K, a type whose terms correspond to subfields of K. Note we do not require the "subfields" to be commutative, so they are really sub-division rings / skew fields. This is the preferred way to talk about subfields in mathlib. Unbundled subfields (s : Set K and IsSubfield 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 K to Subfield K, sending a subset of K to the subfield it generates, and prove that it is a Galois insertion.

Main definitions

Notation used here:

(K : Type u) [DivisionRing K] (L : Type u) [DivisionRing L] (f g : K →+* L) (A : Subfield K) (B : Subfield L) (s : Set K)

• instance : CompleteLattice (Subfield K) : 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 (↑) : Subfield M → Set M form a GaloisInsertion.

• 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.

• f.fieldRange : Subfield L : the range of the ring homomorphism f.

• eqLocusField 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 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

theorem Subfield.list_prod_mem {K : Type u} [DivisionRing K] (s : Subfield K) {l : List K} : (∀ x ∈ l, x ∈ s) → l.prod ∈ s

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

theorem Subfield.list_sum_mem {K : Type u} [DivisionRing K] (s : Subfield K) {l : List K} : (∀ x ∈ l, x ∈ s) → l.sum ∈ s

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

theorem Subfield.multiset_sum_mem {K : Type u} [DivisionRing K] (s : Subfield K) (m : Multiset K) : (∀ a ∈ m, a ∈ s) → m.sum ∈ s

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

theorem Subfield.sum_mem {K : Type u} [DivisionRing K] (s : Subfield K) {ι : Type u_1} {t : Finset ι} {f : ι → K} (h : ∀ c ∈ t, f c ∈ s) : ∑ i ∈ t, f i ∈ s

Sum of elements in a Subfield indexed by a Finset is in the Subfield.

top

@[implicit_reducible]

instance Subfield.instTop {K : Type u} [DivisionRing K] : Top (Subfield K)

The subfield of K containing all elements of K.

Equations

• Subfield.instTop = { top := let __src := ⊤; { toSubring := __src, inv_mem' := ⋯ } }

@[implicit_reducible]

instance Subfield.instInhabited {K : Type u} [DivisionRing K] : Inhabited (Subfield K)

Equations

• Subfield.instInhabited = { default := ⊤ }

@[simp]

theorem Subfield.mem_top {K : Type u} [DivisionRing K] (x : K) : x ∈ ⊤

@[simp]

theorem Subfield.coe_top {K : Type u} [DivisionRing K] : ↑⊤ = Set.univ

def Subfield.topEquiv {K : Type u} [DivisionRing K] : ↥⊤ ≃+* K

The ring equiv between the top element of Subfield K and K.

Equations

• Subfield.topEquiv = Subsemiring.topEquiv

comap

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

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

Equations

• Subfield.comap f s = { toSubring := Subring.comap f s.toSubring, inv_mem' := ⋯ }

@[simp]

theorem Subfield.coe_comap {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) (s : Subfield L) : ↑(comap f s) = ⇑f ⁻¹' ↑s

@[simp]

theorem Subfield.mem_comap {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] {s : Subfield L} {f : K →+* L} {x : K} : x ∈ comap f s ↔ f x ∈ s

theorem Subfield.comap_comap {K : Type u} {L : Type v} {M : Type w} [DivisionRing K] [DivisionRing L] [DivisionRing M] (s : Subfield M) (g : L →+* M) (f : K →+* L) : comap f (comap g s) = comap (g.comp f) s

map

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

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

Equations

• Subfield.map f s = { toSubring := Subring.map f s.toSubring, inv_mem' := ⋯ }

@[simp]

theorem Subfield.coe_map {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (s : Subfield K) (f : K →+* L) : ↑(map f s) = ⇑f '' ↑s

@[simp]

theorem Subfield.mem_map {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] {f : K →+* L} {s : Subfield K} {y : L} : y ∈ map f s ↔ ∃ x ∈ s, f x = y

@[simp]

theorem Subfield.map_mem_map {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) {s : Subfield K} {x : K} : f x ∈ map f s ↔ x ∈ s

theorem Subfield.map_map {K : Type u} {L : Type v} {M : Type w} [DivisionRing K] [DivisionRing L] [DivisionRing M] (s : Subfield K) (g : L →+* M) (f : K →+* L) : map g (map f s) = map (g.comp f) s

theorem Subfield.map_le_iff_le_comap {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] {f : K →+* L} {s : Subfield K} {t : Subfield L} : map f s ≤ t ↔ s ≤ comap f t

theorem Subfield.gc_map_comap {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) : GaloisConnection (map f) (comap f)

range

def RingHom.fieldRange {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) : Subfield L

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

Equations

• f.fieldRange = (Subfield.map f ⊤).copy (Set.range ⇑f) ⋯

@[simp]

theorem RingHom.coe_fieldRange {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) : ↑f.fieldRange = Set.range ⇑f

@[simp]

theorem RingHom.mem_fieldRange {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] {f : K →+* L} {y : L} : y ∈ f.fieldRange ↔ ∃ (x : K), f x = y

theorem RingHom.fieldRange_eq_map {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) : f.fieldRange = Subfield.map f ⊤

theorem RingHom.map_fieldRange {K : Type u} {L : Type v} {M : Type w} [DivisionRing K] [DivisionRing L] [DivisionRing M] (g : L →+* M) (f : K →+* L) : Subfield.map g f.fieldRange = (g.comp f).fieldRange

theorem RingHom.mem_fieldRange_self {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) (x : K) : f x ∈ f.fieldRange

theorem RingHom.fieldRange_eq_top_iff {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] {f : K →+* L} : f.fieldRange = ⊤ ↔ Function.Surjective ⇑f

@[implicit_reducible]

instance RingHom.fintypeFieldRange {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] [Fintype K] [DecidableEq L] (f : K →+* L) : Fintype ↥f.fieldRange

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.fintypeFieldRange = Set.fintypeRange ⇑f

inf

@[implicit_reducible]

instance Subfield.instMin {K : Type u} [DivisionRing K] : Min (Subfield K)

The inf of two subfields is their intersection.

Equations

• Subfield.instMin = { min := fun (s t : Subfield K) => let __src := s.toSubring ⊓ t.toSubring; { toSubring := __src, inv_mem' := ⋯ } }

@[simp]

theorem Subfield.coe_inf {K : Type u} [DivisionRing K] (p p' : Subfield K) : ↑(p ⊓ p') = p.carrier ∩ p'.carrier

@[simp]

theorem Subfield.mem_inf {K : Type u} [DivisionRing K] {p p' : Subfield K} {x : K} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

@[implicit_reducible]

instance Subfield.instInfSet {K : Type u} [DivisionRing K] : InfSet (Subfield K)

Equations

• Subfield.instInfSet = { sInf := fun (S : Set (Subfield K)) => let __src := sInf (Subfield.toSubring '' S); { toSubring := __src, inv_mem' := ⋯ } }

@[simp]

theorem Subfield.coe_sInf {K : Type u} [DivisionRing K] (S : Set (Subfield K)) : ↑(sInf S) = ⋂ s ∈ S, ↑s

@[simp]

theorem Subfield.mem_sInf {K : Type u} [DivisionRing K] {S : Set (Subfield K)} {x : K} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

@[simp]

theorem Subfield.coe_iInf {K : Type u} [DivisionRing K] {ι : Sort u_1} {S : ι → Subfield K} : ↑(⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)

@[simp]

theorem Subfield.mem_iInf {K : Type u} [DivisionRing K] {ι : Sort u_1} {S : ι → Subfield K} {x : K} : x ∈ ⨅ (i : ι), S i ↔ ∀ (i : ι), x ∈ S i

@[simp]

theorem Subfield.sInf_toSubring {K : Type u} [DivisionRing K] (s : Set (Subfield K)) : (sInf s).toSubring = ⨅ t ∈ s, t.toSubring

theorem Subfield.isGLB_sInf {K : Type u} [DivisionRing K] (S : Set (Subfield K)) : IsGLB S (sInf S)

@[implicit_reducible]

instance Subfield.instCompleteLattice {K : Type u} [DivisionRing K] : CompleteLattice (Subfield K)

Subfields of a ring form a complete lattice.

Equations

• One or more equations did not get rendered due to their size.

subfield closure of a subset

def Subfield.closure {K : Type u} [DivisionRing K] (s : Set K) : Subfield K

The Subfield generated by a set.

Equations

• Subfield.closure s = sInf {S : Subfield K | s ⊆ ↑S}

theorem Subfield.mem_closure {K : Type u} [DivisionRing K] {x : K} {s : Set K} : x ∈ closure s ↔ ∀ (S : Subfield K), s ⊆ ↑S → x ∈ S

@[simp]

theorem Subfield.subset_closure {K : Type u} [DivisionRing K] {s : Set K} : s ⊆ ↑(closure s)

The subfield generated by a set includes the set.

theorem Subfield.mem_closure_of_mem {K : Type u} [DivisionRing K] {s : Set K} {x : K} (hx : x ∈ s) : x ∈ closure s

theorem Subfield.subring_closure_le {K : Type u} [DivisionRing K] (s : Set K) : Subring.closure s ≤ (closure s).toSubring

theorem Subfield.notMem_of_notMem_closure {K : Type u} [DivisionRing K] {s : Set K} {P : K} (hP : P ∉ closure s) : P ∉ s

@[simp]

theorem Subfield.closure_le {K : Type u} [DivisionRing K] {s : Set K} {t : Subfield K} : 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} [DivisionRing K] ⦃s t : Set K⦄ (h : s ⊆ t) : closure s ≤ 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} [DivisionRing K] {s : Set K} {t : Subfield K} (h₁ : s ⊆ ↑t) (h₂ : t ≤ closure s) : closure s = t

theorem Subfield.closure_induction {K : Type u} [DivisionRing K] {s : Set K} {p : (x : K) → x ∈ closure s → Prop} (mem : ∀ (x : K) (hx : x ∈ s), p x ⋯) (one : p 1 ⋯) (add : ∀ (x y : K) (hx : x ∈ closure s) (hy : y ∈ closure s), p x hx → p y hy → p (x + y) ⋯) (neg : ∀ (x : K) (hx : x ∈ closure s), p x hx → p (-x) ⋯) (inv : ∀ (x : K) (hx : x ∈ closure s), p x hx → p x⁻¹ ⋯) (mul : ∀ (x y : K) (hx : x ∈ closure s) (hy : y ∈ closure s), p x hx → p y hy → p (x * y) ⋯) {x : K} (h : x ∈ closure s) : p x h

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.

def Subfield.gi (K : Type u) [DivisionRing K] : GaloisInsertion closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations

• Subfield.gi K = { choice := fun (s : Set K) (x : ↑(Subfield.closure s) ≤ s) => Subfield.closure s, gc := ⋯, le_l_u := ⋯, choice_eq := ⋯ }

@[simp]

theorem Subfield.closure_eq {K : Type u} [DivisionRing K] (s : Subfield K) : closure ↑s = s

Closure of a subfield S equals S.

@[simp]

theorem Subfield.closure_empty {K : Type u} [DivisionRing K] : closure ∅ = ⊥

@[simp]

theorem Subfield.closure_univ {K : Type u} [DivisionRing K] : closure Set.univ = ⊤

theorem Subfield.closure_union {K : Type u} [DivisionRing K] (s t : Set K) : closure (s ∪ t) = closure s ⊔ closure t

theorem Subfield.closure_iUnion {K : Type u} [DivisionRing K] {ι : Sort u_1} (s : ι → Set K) : closure (⋃ (i : ι), s i) = ⨆ (i : ι), closure (s i)

theorem Subfield.closure_sUnion {K : Type u} [DivisionRing K] (s : Set (Set K)) : closure (⋃₀ s) = ⨆ t ∈ s, closure t

theorem Subfield.map_sup {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (s t : Subfield K) (f : K →+* L) : map f (s ⊔ t) = map f s ⊔ map f t

theorem Subfield.map_iSup {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] {ι : Sort u_1} (f : K →+* L) (s : ι → Subfield K) : map f (iSup s) = ⨆ (i : ι), map f (s i)

theorem Subfield.map_inf {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (s t : Subfield K) (f : K →+* L) : map f (s ⊓ t) = map f s ⊓ map f t

theorem Subfield.map_iInf {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] {ι : Sort u_1} [Nonempty ι] (f : K →+* L) (s : ι → Subfield K) : map f (iInf s) = ⨅ (i : ι), map f (s i)

theorem Subfield.comap_inf {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (s t : Subfield L) (f : K →+* L) : comap f (s ⊓ t) = comap f s ⊓ comap f t

theorem Subfield.comap_iInf {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] {ι : Sort u_1} (f : K →+* L) (s : ι → Subfield L) : comap f (iInf s) = ⨅ (i : ι), comap f (s i)

@[simp]

theorem Subfield.map_bot {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) : map f ⊥ = ⊥

@[simp]

theorem Subfield.comap_top {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) : comap f ⊤ = ⊤

theorem Subfield.mem_iSup_of_directed {K : Type u} [DivisionRing K] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subfield K} (hS : Directed (fun (x1 x2 : Subfield K) => x1 ≤ x2) S) {x : K} : x ∈ ⨆ (i : ι), S i ↔ ∃ (i : ι), x ∈ S i

The underlying set of a non-empty directed sSup 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_iSup_of_directed {K : Type u} [DivisionRing K] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subfield K} (hS : Directed (fun (x1 x2 : Subfield K) => x1 ≤ x2) S) : ↑(⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

theorem Subfield.mem_sSup_of_directedOn {K : Type u} [DivisionRing K] {S : Set (Subfield K)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : Subfield K) => x1 ≤ x2) S) {x : K} : x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s

theorem Subfield.coe_sSup_of_directedOn {K : Type u} [DivisionRing K] {S : Set (Subfield K)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : Subfield K) => x1 ≤ x2) S) : ↑(sSup S) = ⋃ s ∈ S, ↑s

class Field.FG (L : Type v) [DivisionRing L] : Prop

A field is finitely generated if it is the closure of a finite subset.

• finitely_generated : ∃ (S : Finset L), Subfield.closure ↑S = ⊤

theorem Field.fg_iff (L : Type v) [DivisionRing L] : Field.FG L ↔ ∃ (S : Finset L), Subfield.closure ↑S = ⊤

def RingHom.rangeRestrictField {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) : K →+* ↥f.fieldRange

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

Equations

• f.rangeRestrictField = f.rangeSRestrict

@[simp]

theorem RingHom.coe_rangeRestrictField {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) (x : K) : ↑(f.rangeRestrictField x) = f x

theorem RingHom.rangeRestrictField_bijective {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) : Function.Bijective ⇑f.rangeRestrictField

noncomputable def RingHom.rangeRestrictFieldEquiv {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) : K ≃+* ↥f.fieldRange

RingHom.rangeRestrictField as a RingEquiv.

Equations

• f.rangeRestrictFieldEquiv = RingEquiv.ofBijective f.rangeRestrictField ⋯

@[simp]

theorem RingHom.rangeRestrictFieldEquiv_apply_coe {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) (a : K) : ↑(f.rangeRestrictFieldEquiv a) = f a

@[simp]

theorem RingHom.rangeRestrictFieldEquiv_apply_symm_apply {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) (x : ↥f.fieldRange) : f (f.rangeRestrictFieldEquiv.symm x) = ↑x

def RingHom.eqLocusField {K : Type u} [DivisionRing K] {L : Type v} [Semiring L] (f g : K →+* L) : Subfield K

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.eqLocusField g = { carrier := {x : K | f x = g x}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯, inv_mem' := ⋯ }

@[simp]

theorem RingHom.mem_eqLocusField {K : Type u} [DivisionRing K] {L : Type v} [Semiring L] {f g : K →+* L} {x : K} : x ∈ f.eqLocusField g ↔ f x = g x

theorem RingHom.eqOn_field_closure {K : Type u} [DivisionRing K] {L : Type v} [Semiring L] {f g : K →+* L} {s : Set K} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Subfield.closure s)

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

theorem RingHom.eq_of_eqOn_subfield_top {K : Type u} [DivisionRing K] {L : Type v} [Semiring L] {f g : K →+* L} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g

theorem RingHom.eq_of_eqOn_of_field_closure_eq_top {K : Type u} [DivisionRing K] {L : Type v} [Semiring L] {s : Set K} (hs : Subfield.closure s = ⊤) {f g : K →+* L} (h : Set.EqOn (⇑f) (⇑g) s) : f = g

theorem RingHom.field_closure_preimage_le {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) (s : Set L) : Subfield.closure (⇑f ⁻¹' s) ≤ Subfield.comap f (Subfield.closure s)

theorem RingHom.map_field_closure {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing 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} [DivisionRing 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.codRestrict T ⋯

@[simp]

theorem Subfield.fieldRange_subtype {K : Type u} [DivisionRing K] (s : Subfield K) : s.subtype.fieldRange = s

def RingEquiv.subfieldCongr {K : Type u} [DivisionRing 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

• RingEquiv.subfieldCongr h = { toEquiv := Equiv.setCongr ⋯, map_mul' := ⋯, map_add' := ⋯ }

theorem Subfield.closure_preimage_le {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) (s : Set L) : closure (⇑f ⁻¹' s) ≤ comap f (closure s)

theorem Subfield.multiset_prod_mem {K : Type u} [Field K] (s : Subfield K) (m : Multiset K) : (∀ a ∈ m, a ∈ s) → m.prod ∈ s

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

theorem Subfield.prod_mem {K : Type u} [Field K] (s : Subfield K) {ι : Type u_1} {t : Finset ι} {f : ι → K} (h : ∀ c ∈ t, f c ∈ s) : ∏ i ∈ t, f i ∈ s

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

@[implicit_reducible]

instance Subfield.toAlgebra {K : Type u} [Field K] (s : Subfield K) : Algebra (↥s) K

Equations

• s.toAlgebra = inferInstance

theorem Subfield.algebraMap_ofSubfield {K : Type u} [Field K] (s : Subfield K) : algebraMap (↥s) K = s.subtype

theorem Subfield.mem_closure_iff {K : Type u} [Field K] {s : Set K} {x : K} : x ∈ closure s ↔ ∃ y ∈ Subring.closure s, ∃ z ∈ Subring.closure s, y / z = x

theorem Subfield.map_comap_eq {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) (s : Subfield L) : map f (comap f s) = s ⊓ f.fieldRange

theorem Subfield.map_comap_eq_self {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] {f : K →+* L} {s : Subfield L} (h : s ≤ f.fieldRange) : map f (comap f s) = s

theorem Subfield.map_comap_eq_self_of_surjective {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] {f : K →+* L} (hf : Function.Surjective ⇑f) (s : Subfield L) : map f (comap f s) = s

theorem Subfield.comap_map {K : Type u} {L : Type v} [DivisionRing K] [DivisionRing L] (f : K →+* L) (s : Subfield K) : comap f (map f s) = s

Actions by Subfields

These are just copies of the definitions about Subsemiring starting from Subsemiring.MulAction.

@[implicit_reducible]

instance Subfield.instSMulSubtypeMem {K : Type u} [DivisionRing K] {X : Type u_1} [SMul K X] (F : Subfield K) : SMul (↥F) X

The action by a subfield is the action by the underlying field.

Equations

• F.instSMulSubtypeMem = { smul := fun (m : ↥F.toSubsemiring) (a : X) => ↑m • a }

theorem Subfield.smul_def {K : Type u} [DivisionRing K] {X : Type u_1} [SMul K X] {F : Subfield K} (g : ↥F) (m : X) : g • m = ↑g • m

instance Subfield.smulCommClass_left {K : Type u} [DivisionRing K] {X : Type u_2} {Y : Type u_1} [SMul K Y] [SMul X Y] [SMulCommClass K X Y] (F : Subfield K) : SMulCommClass (↥F) X Y

instance Subfield.smulCommClass_right {K : Type u} [DivisionRing K] {X : Type u_1} {Y : Type u_2} [SMul X Y] [SMul K Y] [SMulCommClass X K Y] (F : Subfield K) : SMulCommClass X (↥F) Y

instance Subfield.instIsScalarTowerSubtypeMem {K : Type u} [DivisionRing K] {X : Type u_1} {Y : Type u_2} [SMul X Y] [SMul K X] [SMul K Y] [IsScalarTower K X Y] (F : Subfield K) : IsScalarTower (↥F) X Y

Note that this provides IsScalarTower F K K which is needed by smul_mul_assoc.

instance Subfield.instFaithfulSMulSubtypeMem {K : Type u} [DivisionRing K] {X : Type u_1} [SMul K X] [FaithfulSMul K X] (F : Subfield K) : FaithfulSMul (↥F) X

@[implicit_reducible]

instance Subfield.instMulActionSubtypeMem {K : Type u} [DivisionRing K] {X : Type u_1} [MulAction K X] (F : Subfield K) : MulAction (↥F) X

The action by a subfield is the action by the underlying field.

Equations

• F.instMulActionSubtypeMem = { toSemigroupAction := (Submonoid.instMulActionSubtypeMem F).toSemigroupAction, one_smul := ⋯ }

@[implicit_reducible]

instance Subfield.instDistribMulActionSubtypeMem {K : Type u} [DivisionRing K] {X : Type u_1} [AddMonoid X] [DistribMulAction K X] (F : Subfield K) : DistribMulAction (↥F) X

The action by a subfield is the action by the underlying field.

Equations

• F.instDistribMulActionSubtypeMem = { toMulAction := F.instMulActionSubtypeMem, smul_zero := ⋯, smul_add := ⋯ }

@[implicit_reducible]

instance Subfield.instMulDistribMulActionSubtypeMem {K : Type u} [DivisionRing K] {X : Type u_1} [Monoid X] [MulDistribMulAction K X] (F : Subfield K) : MulDistribMulAction (↥F) X

The action by a subfield is the action by the underlying field.

Equations

• F.instMulDistribMulActionSubtypeMem = { toMulAction := F.instMulActionSubtypeMem, smul_one := ⋯, smul_mul := ⋯ }

@[implicit_reducible]

instance Subfield.instSMulWithZeroSubtypeMem {K : Type u} [DivisionRing K] {X : Type u_1} [Zero X] [SMulWithZero K X] (F : Subfield K) : SMulWithZero (↥F) X

The action by a subfield is the action by the underlying field.

Equations

• F.instSMulWithZeroSubtypeMem = { toSMulZeroClass := (Subsemiring.instSMulWithZeroSubtypeMem F).toSMulZeroClass, zero_smul := ⋯ }

@[implicit_reducible]

instance Subfield.instMulActionWithZeroSubtypeMem {K : Type u} [DivisionRing K] {X : Type u_1} [Zero X] [MulActionWithZero K X] (F : Subfield K) : MulActionWithZero (↥F) X

The action by a subfield is the action by the underlying field.

Equations

• F.instMulActionWithZeroSubtypeMem = { toMulAction := F.instMulActionSubtypeMem, smul_zero := ⋯, zero_smul := ⋯ }

@[implicit_reducible]

instance Subfield.instModuleSubtypeMem {K : Type u} [DivisionRing K] {X : Type u_1} [AddCommMonoid X] [Module K X] (F : Subfield K) : Module (↥F) X

The action by a subfield is the action by the underlying field.

Equations

• F.instModuleSubtypeMem = { toDistribMulAction := F.instDistribMulActionSubtypeMem, add_smul := ⋯, zero_smul := ⋯ }

@[implicit_reducible]

instance Subfield.instMulSemiringActionSubtypeMem {K : Type u} [DivisionRing K] {X : Type u_1} [Semiring X] [MulSemiringAction K X] (F : Subfield K) : MulSemiringAction (↥F) X

The action by a subfield is the action by the underlying field.

Equations

• F.instMulSemiringActionSubtypeMem = { toDistribMulAction := F.instDistribMulActionSubtypeMem, smul_one := ⋯, smul_mul := ⋯ }