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 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 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 : CompleteLattice (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 (↑) : 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.

• prod A B : Subfield (K × L) : the product of subfields

• f.fieldRange : Subfield B : 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

class SubfieldClass (S : Type u_1) (K : Type u_2) [Field K] [SetLike S K] extends SubringClass , InvMemClass : Prop

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

instance SubfieldClass.toSubgroupClass {K : Type u} [Field K] (S : Type u_1) [SetLike S K] [h : SubfieldClass S K] : SubgroupClass S K

A subfield contains 1, products and inverses.

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

Equations

• @SubfieldClass.toSubgroupClass = @SubfieldClass.toSubgroupClass.proof_1

theorem SubfieldClass.coe_rat_mem {K : Type u} [Field K] {S : Type u_1} [SetLike S K] [h : SubfieldClass S K] (s : S) (x : ℚ) : ↑x ∈ s

instance SubfieldClass.instRatCastSubtypeMemInstMembership {K : Type u} [Field K] {S : Type u_1} [SetLike S K] [h : SubfieldClass S K] (s : S) : RatCast { x // x ∈ s }

Equations

• SubfieldClass.instRatCastSubtypeMemInstMembership s = { ratCast := fun x => { val := ↑x, property := (_ : ↑x ∈ s) } }

@[simp]

theorem SubfieldClass.coe_rat_cast {K : Type u} [Field K] {S : Type u_1} [SetLike S K] [h : SubfieldClass S K] (s : S) (x : ℚ) : ↑↑x = ↑x

theorem SubfieldClass.rat_smul_mem {K : Type u} [Field K] {S : Type u_1} [SetLike S K] [h : SubfieldClass S K] (s : S) (a : ℚ) (x : { x // x ∈ s }) : a • ↑x ∈ s

instance SubfieldClass.instSMulRatSubtypeMemInstMembership {K : Type u} [Field K] {S : Type u_1} [SetLike S K] [h : SubfieldClass S K] (s : S) : SMul ℚ { x // x ∈ s }

Equations

• SubfieldClass.instSMulRatSubtypeMemInstMembership s = { smul := fun a x => { val := a • ↑x, property := (_ : a • ↑x ∈ s) } }

@[simp]

theorem SubfieldClass.coe_rat_smul {K : Type u} [Field K] {S : Type u_1} [SetLike S K] [h : SubfieldClass S K] (s : S) (a : ℚ) (x : { x // x ∈ s }) : ↑(a • x) = a • ↑x

instance SubfieldClass.toField {K : Type u} [Field K] (S : Type u_1) [SetLike S K] [h : SubfieldClass S K] (s : S) : Field { x // x ∈ s }

A subfield inherits a field structure

Equations

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

instance SubfieldClass.toLinearOrderedField (S : Type u_1) {K : Type u_2} [LinearOrderedField K] [SetLike S K] [SubfieldClass S K] (s : S) : LinearOrderedField { x // x ∈ s }

A subfield of a LinearOrderedField is a LinearOrderedField.

Equations

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

structure Subfield (K : Type u) [Field K] extends Subring : Type u

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

  A subfield is closed under multiplicative inverses.

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.toAddSubgroup {K : Type u} [Field K] (s : Subfield K) : AddSubgroup K

The underlying AddSubgroup of a subfield.

Equations

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

instance Subfield.instSetLikeSubfield {K : Type u} [Field K] : SetLike (Subfield K) K

Equations

• Subfield.instSetLikeSubfield = { coe := fun s => s.carrier, coe_injective' := (_ : ∀ (p q : Subfield K), (fun s => s.carrier) p = (fun s => s.carrier) q → p = q) }

instance Subfield.instSubfieldClassSubfieldInstSetLikeSubfield {K : Type u} [Field K] : SubfieldClass (Subfield K) K

Equations

• @Subfield.instSubfieldClassSubfieldInstSetLikeSubfield = @Subfield.instSubfieldClassSubfieldInstSetLikeSubfield.proof_1

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 : Subring K} {x : K} (h : ∀ (x : K), x ∈ S.carrier → x⁻¹ ∈ S.carrier) : x ∈ { toSubring := S, inv_mem' := h } ↔ x ∈ S

@[simp]

theorem Subfield.coe_set_mk {K : Type u} [Field K] (S : Subring K) (h : ∀ (x : K), x ∈ S.carrier → x⁻¹ ∈ S.carrier) : ↑{ toSubring := S, inv_mem' := h } = ↑S

@[simp]

theorem Subfield.mk_le_mk {K : Type u} [Field K] {S : Subring K} {S' : Subring K} (h : ∀ (x : K), x ∈ S.carrier → x⁻¹ ∈ S.carrier) (h' : ∀ (x : K), x ∈ S'.carrier → x⁻¹ ∈ S'.carrier) : { toSubring := S, inv_mem' := h } ≤ { toSubring := S', inv_mem' := h' } ↔ S ≤ S'

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

Two subfields are equal if they have the same elements.

def Subfield.copy {K : Type u} [Field K] (S : Subfield K) (s : Set K) (hs : s = ↑S) : Subfield K

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

Equations

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

@[simp]

theorem Subfield.coe_copy {K : Type u} [Field K] (S : Subfield K) (s : Set K) (hs : s = ↑S) : ↑(Subfield.copy S s hs) = s

theorem Subfield.copy_eq {K : Type u} [Field K] (S : Subfield K) (s : Set K) (hs : s = ↑S) : Subfield.copy S s hs = S

@[simp]

theorem Subfield.coe_toSubring {K : Type u} [Field K] (s : Subfield K) : ↑s.toSubring = ↑s

@[simp]

theorem Subfield.mem_toSubring {K : Type u} [Field K] (s : Subfield K) (x : K) : x ∈ s.toSubring ↔ x ∈ s

def Subring.toSubfield {K : Type u} [Field K] (s : Subring K) (hinv : ∀ (x : K), x ∈ s → x⁻¹ ∈ s) : Subfield K

A Subring containing inverses is a Subfield.

Equations

• Subring.toSubfield s hinv = { toSubring := { toSubsemiring := s.toSubsemiring, neg_mem' := (_ : ∀ {x : K}, x ∈ s.carrier → -x ∈ s.carrier) }, inv_mem' := hinv }

theorem Subfield.one_mem {K : Type u} [Field K] (s : Subfield K) : 1 ∈ s

A subfield contains the field's 1.

theorem Subfield.zero_mem {K : Type u} [Field K] (s : Subfield K) : 0 ∈ s

A subfield contains the field's 0.

theorem Subfield.mul_mem {K : Type u} [Field K] (s : Subfield K) {x : K} {y : K} : x ∈ s → y ∈ s → x * y ∈ s

A subfield is closed under multiplication.

theorem Subfield.add_mem {K : Type u} [Field K] (s : Subfield K) {x : K} {y : K} : x ∈ s → y ∈ s → x + y ∈ s

A subfield is closed under addition.

theorem Subfield.neg_mem {K : Type u} [Field K] (s : Subfield K) {x : K} : x ∈ s → -x ∈ s

A subfield is closed under negation.

theorem Subfield.sub_mem {K : Type u} [Field K] (s : Subfield K) {x : K} {y : K} : x ∈ s → y ∈ s → x - y ∈ s

A subfield is closed under subtraction.

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.

theorem Subfield.div_mem {K : Type u} [Field K] (s : Subfield K) {x : K} {y : K} : x ∈ s → y ∈ s → x / y ∈ s

A subfield is closed under division.

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

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

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

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

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

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

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

Sum 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 : ι), c ∈ t → f c ∈ s) : (Finset.prod t fun i => f i) ∈ s

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

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) : (Finset.sum t fun i => f i) ∈ s

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

theorem Subfield.pow_mem {K : Type u} [Field K] (s : Subfield K) {x : K} (hx : x ∈ s) (n : ℕ) : x ^ n ∈ s

theorem Subfield.zsmul_mem {K : Type u} [Field K] (s : Subfield K) {x : K} (hx : x ∈ s) (n : ℤ) : n • x ∈ s

theorem Subfield.coe_int_mem {K : Type u} [Field K] (s : Subfield K) (n : ℤ) : ↑n ∈ s

theorem Subfield.zpow_mem {K : Type u} [Field K] (s : Subfield K) {x : K} (hx : x ∈ s) (n : ℤ) : x ^ n ∈ s

instance Subfield.instRingSubtypeMemSubfieldInstMembershipInstSetLikeSubfield {K : Type u} [Field K] (s : Subfield K) : Ring { x // x ∈ s }

Equations

• Subfield.instRingSubtypeMemSubfieldInstMembershipInstSetLikeSubfield s = Subring.toRing s.toSubring

instance Subfield.instDivSubtypeMemSubfieldInstMembershipInstSetLikeSubfield {K : Type u} [Field K] (s : Subfield K) : Div { x // x ∈ s }

Equations

• Subfield.instDivSubtypeMemSubfieldInstMembershipInstSetLikeSubfield s = { div := fun x y => { val := ↑x / ↑y, property := (_ : ↑x / ↑y ∈ s) } }

instance Subfield.instInvSubtypeMemSubfieldInstMembershipInstSetLikeSubfield {K : Type u} [Field K] (s : Subfield K) : Inv { x // x ∈ s }

Equations

• Subfield.instInvSubtypeMemSubfieldInstMembershipInstSetLikeSubfield s = { inv := fun x => { val := (↑x)⁻¹, property := (_ : (↑x)⁻¹ ∈ s) } }

instance Subfield.instPowSubtypeMemSubfieldInstMembershipInstSetLikeSubfieldInt {K : Type u} [Field K] (s : Subfield K) : Pow { x // x ∈ s } ℤ

Equations

• Subfield.instPowSubtypeMemSubfieldInstMembershipInstSetLikeSubfieldInt s = { pow := fun x z => { val := ↑(x ^ z), property := (_ : ↑x ^ z ∈ s) } }

instance Subfield.toField {K : Type u} [Field K] (s : Subfield K) : Field { x // x ∈ s }

A subfield inherits a field structure

Equations

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

instance Subfield.toLinearOrderedField {K : Type u_1} [LinearOrderedField K] (s : Subfield K) : LinearOrderedField { x // x ∈ s }

A subfield of a LinearOrderedField is a LinearOrderedField.

Equations

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

@[simp]

theorem Subfield.coe_add {K : Type u} [Field K] (s : Subfield K) (x : { x // x ∈ s }) (y : { x // x ∈ s }) : ↑(x + y) = ↑x + ↑y

@[simp]

theorem Subfield.coe_sub {K : Type u} [Field K] (s : Subfield K) (x : { x // x ∈ s }) (y : { x // x ∈ s }) : ↑(x - y) = ↑x - ↑y

@[simp]

theorem Subfield.coe_neg {K : Type u} [Field K] (s : Subfield K) (x : { x // x ∈ s }) : ↑(-x) = -↑x

@[simp]

theorem Subfield.coe_mul {K : Type u} [Field K] (s : Subfield K) (x : { x // x ∈ s }) (y : { x // x ∈ s }) : ↑(x * y) = ↑x * ↑y

@[simp]

theorem Subfield.coe_div {K : Type u} [Field K] (s : Subfield K) (x : { x // x ∈ s }) (y : { x // x ∈ s }) : ↑(x / y) = ↑x / ↑y

@[simp]

theorem Subfield.coe_inv {K : Type u} [Field K] (s : Subfield K) (x : { x // x ∈ s }) : ↑x⁻¹ = (↑x)⁻¹

@[simp]

theorem Subfield.coe_zero {K : Type u} [Field K] (s : Subfield K) : ↑0 = 0

@[simp]

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) : { x // x ∈ s } →+* K

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

Equations

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

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

Equations

• Subfield.toAlgebra s = RingHom.toAlgebra (Subfield.subtype s)

@[simp]

theorem Subfield.coe_subtype {K : Type u} [Field K] (s : Subfield K) : ↑(Subfield.subtype s) = Subtype.val

theorem Subfield.toSubring_subtype_eq_subtype (F : Type u_1) [Field F] (S : Subfield F) : Subring.subtype S.toSubring = Subfield.subtype S

Partial order

theorem Subfield.mem_toSubmonoid {K : Type u} [Field K] {s : Subfield K} {x : K} : x ∈ s.toSubmonoid ↔ x ∈ s

@[simp]

theorem Subfield.coe_toSubmonoid {K : Type u} [Field K] (s : Subfield K) : ↑s.toSubmonoid = ↑s

@[simp]

theorem Subfield.mem_toAddSubgroup {K : Type u} [Field K] {s : Subfield K} {x : K} : x ∈ Subfield.toAddSubgroup s ↔ x ∈ s

@[simp]

theorem Subfield.coe_toAddSubgroup {K : Type u} [Field K] (s : Subfield K) : ↑(Subfield.toAddSubgroup s) = ↑s

top

instance Subfield.instTopSubfield {K : Type u} [Field K] : Top (Subfield K)

The subfield of K containing all elements of K.

Equations

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

instance Subfield.instInhabitedSubfield {K : Type u} [Field K] : Inhabited (Subfield K)

Equations

• Subfield.instInhabitedSubfield = { default := ⊤ }

@[simp]

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

@[simp]

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

@[simp]

theorem Subfield.topEquiv_apply {K : Type u} [Field K] (r : { x // x ∈ ⊤ }) : ↑Subfield.topEquiv r = ↑r

@[simp]

theorem Subfield.topEquiv_symm_apply_coe {K : Type u} [Field K] (r : K) : ↑(↑(RingEquiv.symm Subfield.topEquiv) r) = r

def Subfield.topEquiv {K : Type u} [Field K] : { x // x ∈ ⊤ } ≃+* 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} [Field K] [Field L] (f : K →+* L) (s : Subfield L) : Subfield K

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

Equations

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

@[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 (RingHom.comp g f) s

map

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

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

Equations

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

@[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, 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 (RingHom.comp g 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) : GaloisConnection (Subfield.map f) (Subfield.comap f)

range

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

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

Equations

• RingHom.fieldRange f = Subfield.copy (Subfield.map f ⊤) (Set.range ↑f) (_ : Set.range ↑f = ↑f '' Set.univ)

@[simp]

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

@[simp]

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

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

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

instance RingHom.fintypeFieldRange {K : Type u} {L : Type v} [Field K] [Field L] [Fintype K] [DecidableEq L] (f : K →+* L) : Fintype { x // x ∈ RingHom.fieldRange f }

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

• RingHom.fintypeFieldRange f = Set.fintypeRange ↑f

inf

instance Subfield.instInfSubfield {K : Type u} [Field K] : Inf (Subfield K)

The inf of two subfields is their intersection.

Equations

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

@[simp]

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

@[simp]

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

instance Subfield.instInfSetSubfield {K : Type u} [Field K] : InfSet (Subfield K)

Equations

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

@[simp]

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

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

@[simp]

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

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

instance Subfield.instCompleteLatticeSubfield {K : Type u} [Field 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} [Field K] (s : Set K) : Subfield K

The Subfield generated by a set.

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

@[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 : Set K⦄ ⦃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.

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

closure forms a Galois insertion with the coercion to set.

Equations

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

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 : Set K) (t : Set K) : Subfield.closure (s ∪ t) = Subfield.closure s ⊔ Subfield.closure t

theorem Subfield.closure_iUnion {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 (⋃₀ s) = ⨆ (t : Set K) (_ : t ∈ s), Subfield.closure t

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

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

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

theorem Subfield.comap_iInf {K : Type u} {L : Type v} [Field K] [Field L] {ι : Sort u_1} (f : K →+* L) (s : ι → Subfield L) : Subfield.comap f (iInf 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_iSup_of_directed {K : Type u} [Field K] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subfield K} (hS : Directed (fun x x_1 => x ≤ x_1) S) {x : K} : x ∈ ⨆ (i : ι), S i ↔ ∃ i, 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} [Field K] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subfield K} (hS : Directed (fun x x_1 => x ≤ x_1) S) : ↑(⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

theorem Subfield.mem_sSup_of_directedOn {K : Type u} [Field K] {S : Set (Subfield K)} (Sne : Set.Nonempty S) (hS : DirectedOn (fun x x_1 => x ≤ x_1) S) {x : K} : x ∈ sSup S ↔ ∃ s, s ∈ S ∧ x ∈ s

theorem Subfield.coe_sSup_of_directedOn {K : Type u} [Field K] {S : Set (Subfield K)} (Sne : Set.Nonempty S) (hS : DirectedOn (fun x x_1 => x ≤ x_1) S) : ↑(sSup S) = ⋃ (s : Subfield K) (_ : s ∈ S), ↑s

def RingHom.rangeRestrictField {K : Type u} {L : Type v} [Field K] [Field L] (f : K →+* L) : K →+* { x // x ∈ RingHom.fieldRange f }

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

Equations

• RingHom.rangeRestrictField f = RingHom.rangeSRestrict f

@[simp]

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

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

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

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

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

theorem RingHom.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 RingHom.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 : Subfield K} {T : Subfield K} (h : S ≤ T) : { x // x ∈ S } →+* { x // x ∈ T }

The ring homomorphism associated to an inclusion of subfields.

Equations

• Subfield.inclusion h = RingHom.codRestrict (Subfield.subtype S) T (_ : ∀ (x : { x // x ∈ S }), ↑(Subfield.subtype S) x ∈ T)

@[simp]

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

def RingEquiv.subfieldCongr {K : Type u} [Field K] {s : Subfield K} {t : Subfield K} (h : s = t) : { x // x ∈ s } ≃+* { x // x ∈ t }

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

Equations

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

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)