P. 344 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34301-34400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dimkerim 34301	Given a linear map 𝐹 between vector spaces 𝑉 and 𝑈, the dimension of the vector space 𝑉 is the sum of the dimension of 𝐹 's kernel and the dimension of 𝐹's image. Second part of theorem 5.3 in [Lang] p. 141 This can also be described as the Rank-nullity theorem, (dim‘𝐼) being the rank of 𝐹 (the dimension of its image), and (dim‘𝐾) its nullity (the dimension of its kernel). (Contributed by Thierry Arnoux, 17-May-2023.)
⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 }))    &   ⊢ 𝐼 = (𝑈 ↾s ran 𝐹)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +e (dim‘𝐼)))
Theorem	qusdimsum 34302	Let 𝑊 be a vector space, and let 𝑋 be a subspace. Then the dimension of 𝑊 is the sum of the dimension of 𝑋 and the dimension of the quotient space of 𝑋. First part of theorem 5.3 in [Lang] p. 141. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑌 = (𝑊 /s (𝑊 ~QG 𝑈))    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑊) = ((dim‘𝑋) +e (dim‘𝑌)))
Theorem	fedgmullem1 34303*	Lemma for fedgmul 34305. (Contributed by Thierry Arnoux, 20-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉)    &   ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    &   ⊢ 𝐾 = (𝐸 ↾s 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹))    &   ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗))    &   ⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))    &   ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵))    &   ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴))    &   ⊢ (𝜑 → 𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))    &   ⊢ (𝜑 → 𝐿 finSupp (0g‘(Scalar‘𝐵)))    &   ⊢ (𝜑 → 𝑍 = (𝐵 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))))    &   ⊢ (𝜑 → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)))    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐿‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))    ⇒   ⊢ (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷))))
Theorem	fedgmullem2 34304*	Lemma for fedgmul 34305. (Contributed by Thierry Arnoux, 20-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉)    &   ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    &   ⊢ 𝐾 = (𝐸 ↾s 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹))    &   ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗))    &   ⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))    &   ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵))    &   ⊢ (𝜑 → 𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))))    &   ⊢ (𝜑 → (𝐴 Σg (𝑊 ∘f ( ·𝑠 ‘𝐴)𝐷)) = (0g‘𝐴))    ⇒   ⊢ (𝜑 → 𝑊 = ((𝑌 × 𝑋) × {(0g‘(Scalar‘𝐴))}))
Theorem	fedgmul 34305	The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, we have [𝐸:𝐾] = [𝐸:𝐹][𝐹:𝐾]. Proposition 1.2 of [Lang], p. 224. Here (dim‘𝐴) is the degree of the extension 𝐸 of 𝐾, (dim‘𝐵) is the degree of the extension 𝐸 of 𝐹, and (dim‘𝐶) is the degree of the extension 𝐹 of 𝐾. This proof is valid for infinite dimensions, and is actually valid for division ring extensions, not just field extensions. (Contributed by Thierry Arnoux, 25-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉)    &   ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    &   ⊢ 𝐾 = (𝐸 ↾s 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹))    ⇒   ⊢ (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
Theorem	dimlssid 34306	If the dimension of a linear subspace 𝐿 is the dimension of the whole vector space 𝐸, then 𝐿 is the whole space. (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ LVec)    &   ⊢ (𝜑 → (dim‘𝐸) ∈ ℕ0)    &   ⊢ (𝜑 → 𝐿 ∈ (LSubSp‘𝐸))    &   ⊢ (𝜑 → (dim‘(𝐸 ↾s 𝐿)) = (dim‘𝐸))    ⇒   ⊢ (𝜑 → 𝐿 = 𝐵)
Theorem	lvecendof1f1o 34307	If an endomorphism 𝑈 of a vector space 𝐸 of finite dimension is injective, then it is bijective. Item (b) of Corollary of Proposition 9 in [BourbakiAlg1] p. 298 . (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ LVec)    &   ⊢ (𝜑 → (dim‘𝐸) ∈ ℕ0)    &   ⊢ (𝜑 → 𝑈 ∈ (𝐸 LMHom 𝐸))    &   ⊢ (𝜑 → 𝑈:𝐵–1-1→𝐵)    ⇒   ⊢ (𝜑 → 𝑈:𝐵–1-1-onto→𝐵)
Theorem	lactlmhm 34308*	In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is a module homomorphism, analogous to ringlghm 20543. (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ · = (.r‘𝐴)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐶 · 𝑥))    &   ⊢ (𝜑 → 𝐴 ∈ AssAlg)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐴 LMHom 𝐴))
Theorem	assalactf1o 34309*	In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 34308. (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ · = (.r‘𝐴)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐶 · 𝑥))    &   ⊢ (𝜑 → 𝐴 ∈ AssAlg)    &   ⊢ 𝐸 = (RLReg‘𝐴)    &   ⊢ 𝐾 = (Scalar‘𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → (dim‘𝐴) ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐸)    ⇒   ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐵)
Theorem	assarrginv 34310	If an element 𝑋 of an associative algebra 𝐴 over a division ring 𝐾 is regular, then it is a unit. Proposition 2. in Chapter 5. of [BourbakiAlg2] p. 113. (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ 𝐸 = (RLReg‘𝐴)    &   ⊢ 𝑈 = (Unit‘𝐴)    &   ⊢ 𝐾 = (Scalar‘𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ AssAlg)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → (dim‘𝐴) ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑈)
Theorem	assafld 34311	If an algebra 𝐴 of finite degree over a division ring 𝐾 is an integral domain, then it is a field. Corollary of Proposition 2. in Chapter 5. of [BourbakiAlg2] p. 113. (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ 𝐾 = (Scalar‘𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ AssAlg)    &   ⊢ (𝜑 → 𝐴 ∈ IDomn)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → (dim‘𝐴) ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ Field)
21.4.11  Field Extensions
Syntax	cfldext 34312	Syntax for the field extension relation.
class /FldExt
Syntax	cfinext 34313	Syntax for the finite field extension relation.
class /FinExt
Syntax	cextdg 34314	Syntax for the field extension degree operation.
class [:]
Definition	df-fldext 34315*	Definition of the field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
Definition	df-extdg 34316*	Definition of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
Definition	df-finext 34317*	Definition of the finite field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
Theorem	relfldext 34318	The field extension is a relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ Rel /FldExt
Theorem	brfldext 34319	The field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
Theorem	ccfldextrr 34320	The field of the complex numbers is an extension of the field of the real numbers. (Contributed by Thierry Arnoux, 20-Jul-2023.)
⊢ ℂfld/FldExtℝfld
Theorem	fldextfld1 34321	A field extension is only defined if the extension is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
Theorem	fldextfld2 34322	A field extension is only defined if the subfield is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
Theorem	fldextsubrg 34323	Field extension implies a subring relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ 𝑈 = (Base‘𝐹)    ⇒   ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸))
Theorem	sdrgfldext 34324	A field 𝐸 and any sub-division-ring 𝐹 of 𝐸 form a field extension. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    ⇒   ⊢ (𝜑 → 𝐸/FldExt(𝐸 ↾s 𝐹))
Theorem	fldextress 34325	Field extension implies a structure restriction relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
Theorem	brfinext 34326	The finite field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0))
Theorem	extdgval 34327	Value of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
Theorem	fldextsdrg 34328	Deduce sub-division-ring from field extension. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐸/FldExt𝐹)    ⇒   ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐸))
Theorem	fldextsralvec 34329	The subring algebra associated with a field extension is a vector space. (Contributed by Thierry Arnoux, 3-Aug-2023.)
⊢ (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
Theorem	extdgcl 34330	Closure of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) ∈ ℕ0*)
Theorem	extdggt0 34331	Degrees of field extension are greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → 0 < (𝐸[:]𝐹))
Theorem	fldexttr 34332	Field extension is a transitive relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
Theorem	fldextid 34333	The field extension relation is reflexive. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ (𝐹 ∈ Field → 𝐹/FldExt𝐹)
Theorem	extdgid 34334	A trivial field extension has degree one. (Contributed by Thierry Arnoux, 4-Aug-2023.)
⊢ (𝐸 ∈ Field → (𝐸[:]𝐸) = 1)
Theorem	fldsdrgfldext 34335	A sub-division-ring of a field forms a field extension. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐺 = (𝐹 ↾s 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ Field)    &   ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝐹))    ⇒   ⊢ (𝜑 → 𝐹/FldExt𝐺)
Theorem	fldsdrgfldext2 34336	A sub-sub-division-ring of a field forms a field extension. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐺 = (𝐹 ↾s 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ Field)    &   ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝐹))    &   ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐺))    &   ⊢ 𝐻 = (𝐹 ↾s 𝐵)    ⇒   ⊢ (𝜑 → 𝐺/FldExt𝐻)
Theorem	extdgmul 34337	The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, the degree of the extension 𝐸/FldExt𝐾 is the product of the degrees of the extensions 𝐸/FldExt𝐹 and 𝐹/FldExt𝐾. Proposition 1.2 of [Lang], p. 224. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))
Theorem	finextfldext 34338	A finite field extension is a field extension. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐸/FinExt𝐹)    ⇒   ⊢ (𝜑 → 𝐸/FldExt𝐹)
Theorem	finexttrb 34339	The extension 𝐸 of 𝐾 is finite if and only if 𝐸 is finite over 𝐹 and 𝐹 is finite over 𝐾. Corollary 1.3 of [Lang] , p. 225. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FinExt𝐾 ↔ (𝐸/FinExt𝐹 ∧ 𝐹/FinExt𝐾)))
Theorem	extdg1id 34340	If the degree of the extension 𝐸/FldExt𝐹 is 1, then 𝐸 and 𝐹 are identical. (Contributed by Thierry Arnoux, 6-Aug-2023.)
⊢ ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹)
Theorem	extdg1b 34341	The degree of the extension 𝐸/FldExt𝐹 is 1 iff 𝐸 and 𝐹 are the same structure. (Contributed by Thierry Arnoux, 6-Aug-2023.)
⊢ (𝐸/FldExt𝐹 → ((𝐸[:]𝐹) = 1 ↔ 𝐸 = 𝐹))
Theorem	fldgenfldext 34342	A subfield 𝐹 extended with a set 𝐴 forms a field extension. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴)))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐿/FldExt𝐾)
Theorem	fldextchr 34343	The characteristic of a subfield is the same as the characteristic of the larger field. (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ (𝐸/FldExt𝐹 → (chr‘𝐹) = (chr‘𝐸))
Theorem	evls1fldgencl 34344	Closure of the subring polynomial evaluation in the field extention. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
Theorem	ccfldsrarelvec 34345	The subring algebra of the complex numbers over the real numbers is a left vector space. (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ ((subringAlg ‘ℂfld)‘ℝ) ∈ LVec
Theorem	ccfldextdgrr 34346	The degree of the field extension of the complex numbers over the real numbers is 2. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ (ℂfld[:]ℝfld) = 2
Theorem	fldextrspunlsplem 34347*	Lemma for fldextrspunlsp 34348: First direction. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ 𝐾 = (𝐿 ↾s 𝐹)    &   ⊢ 𝐼 = (𝐿 ↾s 𝐺)    &   ⊢ 𝐽 = (𝐿 ↾s 𝐻)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))    &   ⊢ 𝑁 = (RingSpan‘𝐿)    &   ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))    &   ⊢ 𝐸 = (𝐿 ↾s 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝑃:𝐻⟶𝐺)    &   ⊢ (𝜑 → 𝑃 finSupp (0g‘𝐿))    &   ⊢ (𝜑 → 𝑋 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))))    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))))
Theorem	fldextrspunlsp 34348	Lemma for fldextrspunfld 34350. The subring generated by the union of two field extensions 𝐺 and 𝐻 is the vector sub- 𝐺 space generated by a basis 𝐵 of 𝐻. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ 𝐾 = (𝐿 ↾s 𝐹)    &   ⊢ 𝐼 = (𝐿 ↾s 𝐺)    &   ⊢ 𝐽 = (𝐿 ↾s 𝐻)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))    &   ⊢ 𝑁 = (RingSpan‘𝐿)    &   ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))    &   ⊢ 𝐸 = (𝐿 ↾s 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵))
Theorem	fldextrspunlem1 34349	Lemma for fldextrspunfld 34350. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ 𝐾 = (𝐿 ↾s 𝐹)    &   ⊢ 𝐼 = (𝐿 ↾s 𝐺)    &   ⊢ 𝐽 = (𝐿 ↾s 𝐻)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ0)    &   ⊢ 𝑁 = (RingSpan‘𝐿)    &   ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))    &   ⊢ 𝐸 = (𝐿 ↾s 𝐶)    ⇒   ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))
Theorem	fldextrspunfld 34350	The ring generated by the union of two field extensions is a field. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ 𝐾 = (𝐿 ↾s 𝐹)    &   ⊢ 𝐼 = (𝐿 ↾s 𝐺)    &   ⊢ 𝐽 = (𝐿 ↾s 𝐻)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ0)    &   ⊢ 𝑁 = (RingSpan‘𝐿)    &   ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))    &   ⊢ 𝐸 = (𝐿 ↾s 𝐶)    ⇒   ⊢ (𝜑 → 𝐸 ∈ Field)
Theorem	fldextrspunlem2 34351	Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ 𝐾 = (𝐿 ↾s 𝐹)    &   ⊢ 𝐼 = (𝐿 ↾s 𝐺)    &   ⊢ 𝐽 = (𝐿 ↾s 𝐻)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ0)    &   ⊢ 𝑁 = (RingSpan‘𝐿)    &   ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))    &   ⊢ 𝐸 = (𝐿 ↾s 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 = (𝐿 fldGen (𝐺 ∪ 𝐻)))
Theorem	fldextrspundgle 34352	Inequality involving the degree of two different field extensions 𝐼 and 𝐽 of a same field 𝐹. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ 𝐾 = (𝐿 ↾s 𝐹)    &   ⊢ 𝐼 = (𝐿 ↾s 𝐺)    &   ⊢ 𝐽 = (𝐿 ↾s 𝐻)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ0)    &   ⊢ 𝐸 = (𝐿 ↾s (𝐿 fldGen (𝐺 ∪ 𝐻)))    ⇒   ⊢ (𝜑 → (𝐸[:]𝐼) ≤ (𝐽[:]𝐾))
Theorem	fldextrspundglemul 34353	Given two field extensions 𝐼 / 𝐾 and 𝐽 / 𝐾 of the same field 𝐾, 𝐽 / 𝐾 being finite, and the composiste field 𝐸 = 𝐼𝐽, the degree of the extension of the composite field 𝐸 / 𝐾 is at most the product of the field extension degrees of 𝐼 / 𝐾 and 𝐽 / 𝐾. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐾 = (𝐿 ↾s 𝐹)    &   ⊢ 𝐼 = (𝐿 ↾s 𝐺)    &   ⊢ 𝐽 = (𝐿 ↾s 𝐻)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ0)    &   ⊢ 𝐸 = (𝐿 ↾s (𝐿 fldGen (𝐺 ∪ 𝐻)))    ⇒   ⊢ (𝜑 → (𝐸[:]𝐾) ≤ ((𝐼[:]𝐾) ·e (𝐽[:]𝐾)))
Theorem	fldextrspundgdvdslem 34354	Lemma for fldextrspundgdvds 34355. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐾 = (𝐿 ↾s 𝐹)    &   ⊢ 𝐼 = (𝐿 ↾s 𝐺)    &   ⊢ 𝐽 = (𝐿 ↾s 𝐻)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ0)    &   ⊢ 𝐸 = (𝐿 ↾s (𝐿 fldGen (𝐺 ∪ 𝐻)))    &   ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ0)
Theorem	fldextrspundgdvds 34355	Given two finite extensions 𝐼 / 𝐾 and 𝐽 / 𝐾 of the same field 𝐾, the degree of the extension 𝐼 / 𝐾 divides the degree of the extension 𝐸 / 𝐾, 𝐸 being the composite field 𝐼𝐽. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐾 = (𝐿 ↾s 𝐹)    &   ⊢ 𝐼 = (𝐿 ↾s 𝐺)    &   ⊢ 𝐽 = (𝐿 ↾s 𝐻)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ0)    &   ⊢ 𝐸 = (𝐿 ↾s (𝐿 fldGen (𝐺 ∪ 𝐻)))    &   ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐼[:]𝐾) ∥ (𝐸[:]𝐾))
Theorem	fldext2rspun 34356*	Given two field extensions 𝐼 / 𝐾 and 𝐽 / 𝐾, 𝐼 / 𝐾 being a quadratic extension, and the degree of 𝐽 / 𝐾 being a power of 2, the degree of the extension 𝐸 / 𝐾 is a power of 2 , 𝐸 being the composite field 𝐼𝐽. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐾 = (𝐿 ↾s 𝐹)    &   ⊢ 𝐼 = (𝐿 ↾s 𝐺)    &   ⊢ 𝐽 = (𝐿 ↾s 𝐻)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐼[:]𝐾) = 2)    &   ⊢ (𝜑 → (𝐽[:]𝐾) = (2↑𝑁))    &   ⊢ 𝐸 = (𝐿 ↾s (𝐿 fldGen (𝐺 ∪ 𝐻)))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐸[:]𝐾) = (2↑𝑛))
21.4.11.1  Algebraic numbers
Syntax	cirng 34357	Integral subring of a ring.
class IntgRing
Definition	df-irng 34358*	Define the subring of elements of a ring 𝑟 integral over a subset 𝑠. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Thierry Arnoux, 28-Jan-2025.)
⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}))
Theorem	irngval 34359*	The elements of a field 𝑅 integral over a subset 𝑆. In the case of a subfield, those are the algebraic numbers over the field 𝑆 within the field 𝑅. That is, the numbers 𝑋 which are roots of monic polynomials 𝑃(𝑋) with coefficients in 𝑆. (Contributed by Thierry Arnoux, 28-Jan-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))
Theorem	elirng 34360*	Property for an element 𝑋 of a field 𝑅 to be integral over a subring 𝑆. (Contributed by Thierry Arnoux, 28-Jan-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑋) = 0 )))
Theorem	irngss 34361	All elements of a subring 𝑆 are integral over 𝑆. This is only true in the case of a nonzero ring, since there are no integral elements in a zero ring (see 0ringirng 34363). (Contributed by Thierry Arnoux, 28-Jan-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    ⇒   ⊢ (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))
Theorem	irngssv 34362	An integral element is an element of the base set. (Contributed by Thierry Arnoux, 28-Jan-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   ⊢ (𝜑 → (𝑅 IntgRing 𝑆) ⊆ 𝐵)
Theorem	0ringirng 34363	A zero ring 𝑅 has no integral elements. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → ¬ 𝑅 ∈ NzRing)    ⇒   ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∅)
Theorem	irngnzply1lem 34364	In the case of a field 𝐸, a root 𝑋 of some nonzero polynomial 𝑃 with coefficients in a subfield 𝐹 is integral over 𝐹. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑍 = (0g‘(Poly1‘𝐸))    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝑃 ∈ dom 𝑂)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑍)    &   ⊢ (𝜑 → ((𝑂‘𝑃)‘𝑋) = 0 )    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))
Theorem	irngnzply1 34365*	In the case of a field 𝐸, the roots of nonzero polynomials 𝑝 with coefficients in a subfield 𝐹 are exactly the integral elements over 𝐹. Roots of nonzero polynomials are called algebraic numbers, so this shows that in the case of a field, elements integral over 𝐹 are exactly the algebraic numbers. In this formula, dom 𝑂 represents the polynomials, and 𝑍 the zero polynomial. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑍 = (0g‘(Poly1‘𝐸))    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    ⇒   ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))
Theorem	extdgfialglem1 34366*	Lemma for extdgfialg 34368. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ 𝑍 = (0g‘𝐸)    &   ⊢ · = (.r‘𝐸)    &   ⊢ 𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ (𝐹 ↑m (0...𝐷))(𝑎 finSupp 𝑍 ∧ ((𝐸 Σg (𝑎 ∘f · 𝐺)) = 𝑍 ∧ 𝑎 ≠ ((0...𝐷) × {𝑍}))))
Theorem	extdgfialglem2 34367*	Lemma for extdgfialg 34368. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ 𝑍 = (0g‘𝐸)    &   ⊢ · = (.r‘𝐸)    &   ⊢ 𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴:(0...𝐷)⟶𝐹)    &   ⊢ (𝜑 → 𝐴 finSupp 𝑍)    &   ⊢ (𝜑 → (𝐸 Σg (𝐴 ∘f · 𝐺)) = 𝑍)    &   ⊢ (𝜑 → 𝐴 ≠ ((0...𝐷) × {𝑍}))    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))
Theorem	extdgfialg 34368	A finite field extension 𝐸 / 𝐹 is algebraic. Part of the proof of Proposition 1.1 of [Lang], p. 224. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = 𝐵)
21.4.11.2  Algebraic extensions
Syntax	calgext 34369	Syntax for the algebraic field extension relation.
class /AlgExt
Definition	df-algext 34370*	Definition of the algebraic extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ /AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒))}
Theorem	bralgext 34371	Express the fact that a field extension 𝐸 / 𝐹 is algebraic. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐸/AlgExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))
Theorem	finextalg 34372	A finite field extension is algebraic. Proposition 1.1 of [Lang], p. 224. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐸/FinExt𝐹)    ⇒   ⊢ (𝜑 → 𝐸/AlgExt𝐹)
21.4.11.3  Minimal polynomials
Syntax	cminply 34373	Extend class notation with the minimal polynomial builder function.
class minPoly
Definition	df-minply 34374*	Define the minimal polynomial builder function. (Contributed by Thierry Arnoux, 19-Jan-2025.)
⊢ minPoly = (𝑒 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen1p‘(𝑒 ↾s 𝑓))‘{𝑝 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑝)‘𝑥) = (0g‘𝑒)})))
Theorem	ply1annidllem 34375*	Write the set 𝑄 of polynomials annihilating an element 𝐴 as the kernel of the ring homomorphism 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝐴. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   ⊢ 𝐹 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))    ⇒   ⊢ (𝜑 → 𝑄 = (◡𝐹 “ { 0 }))
Theorem	ply1annidl 34376*	The set 𝑄 of polynomials annihilating an element 𝐴 forms an ideal. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    ⇒   ⊢ (𝜑 → 𝑄 ∈ (LIdeal‘𝑃))
Theorem	ply1annnr 34377*	The set 𝑄 of polynomials annihilating an element 𝐴 is not the whole polynomial ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    ⇒   ⊢ (𝜑 → 𝑄 ≠ 𝑈)
Theorem	ply1annig1p 34378*	The ideal 𝑄 of polynomials annihilating an element 𝐴 is generated by the ideal's canonical generator. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   ⊢ 𝐾 = (RSpan‘𝑃)    &   ⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))    ⇒   ⊢ (𝜑 → 𝑄 = (𝐾‘{(𝐺‘𝑄)}))
Theorem	minplyval 34379*	Expand the value of the minimal polynomial (𝑀‘𝐴) for a given element 𝐴. It is defined as the unique monic polynomial of minimal degree which annihilates 𝐴. By ply1annig1p 34378, that polynomial generates the ideal of the annihilators of 𝐴. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   ⊢ 𝐾 = (RSpan‘𝑃)    &   ⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄))
Theorem	minplycl 34380*	The minimal polynomial is a polynomial. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   ⊢ 𝐾 = (RSpan‘𝑃)    &   ⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
Theorem	ply1annprmidl 34381*	The set 𝑄 of polynomials annihilating an element 𝐴 is a prime ideal. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    ⇒   ⊢ (𝜑 → 𝑄 ∈ (PrmIdeal‘𝑃))
Theorem	minplymindeg 34382	The minimal polynomial of 𝐴 is minimal among the nonzero annihilators of 𝐴 with regard to degree. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ 𝐷 = (deg1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → ((𝑂‘𝐻)‘𝐴) = 0 )    &   ⊢ (𝜑 → 𝐻 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐻 ≠ 𝑍)    ⇒   ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ≤ (𝐷‘𝐻))
Theorem	minplyann 34383	The minimal polynomial for 𝐴 annihilates 𝐴. (Contributed by Thierry Arnoux, 25-Apr-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    ⇒   ⊢ (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 )
Theorem	minplyirredlem 34384	Lemma for minplyirred 34385. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)    &   ⊢ (𝜑 → 𝐺 ∈ (Base‘𝑃))    &   ⊢ (𝜑 → 𝐻 ∈ (Base‘𝑃))    &   ⊢ (𝜑 → (𝐺(.r‘𝑃)𝐻) = (𝑀‘𝐴))    &   ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) = (0g‘𝐸))    &   ⊢ (𝜑 → 𝐺 ≠ 𝑍)    &   ⊢ (𝜑 → 𝐻 ≠ 𝑍)    ⇒   ⊢ (𝜑 → 𝐻 ∈ (Unit‘𝑃))
Theorem	minplyirred 34385	A nonzero minimal polynomial is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))
Theorem	irngnminplynz 34386	Integral elements have nonzero minimal polynomials. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑍 = (0g‘(Poly1‘𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)
Theorem	minplym1p 34387	A minimal polynomial is monic. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑍 = (0g‘(Poly1‘𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑈 = (Monic1p‘(𝐸 ↾s 𝐹))    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
Theorem	minplynzm1p 34388	If a minimal polynomial is nonzero, then it is monic. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝑍 = (0g‘(Poly1‘𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)    &   ⊢ 𝑈 = (Monic1p‘(𝐸 ↾s 𝐹))    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
Theorem	minplyelirng 34389	If the minimal polynomial 𝐹 of an element 𝑋 of a field 𝑅 has nonnegative degree, then 𝑋 is integral. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑀 = (𝑅 minPoly 𝑆)    &   ⊢ 𝐷 = (deg1‘(𝑅 ↾s 𝑆))    &   ⊢ (𝜑 → 𝑅 ∈ Field)    &   ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑅 IntgRing 𝑆))
Theorem	irredminply 34390	An irreducible, monic, annihilating polynomial is the minimal polynomial. (Contributed by Thierry Arnoux, 27-Apr-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) = 0 )    &   ⊢ (𝜑 → 𝐺 ∈ (Irred‘𝑃))    &   ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘(𝐸 ↾s 𝐹)))    ⇒   ⊢ (𝜑 → 𝐺 = (𝑀‘𝐴))
Theorem	algextdeglem1 34391	Lemma for algextdeg 34399. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    ⇒   ⊢ (𝜑 → (𝐿 ↾s 𝐹) = 𝐾)
Theorem	algextdeglem2 34392*	Lemma for algextdeg 34399. Both the ring of polynomials 𝑃 and the field 𝐿 generated by 𝐾 and the algebraic element 𝐴 can be considered as modules over the elements of 𝐹. Then, the evaluation map 𝐺, mapping polynomials to their evaluation at 𝐴, is a module homomorphism between those modules. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))
Theorem	algextdeglem3 34393*	Lemma for algextdeg 34399. The quotient 𝑃 / 𝑍 of the vector space 𝑃 of polynomials by the subspace 𝑍 of polynomials annihilating 𝐴 is itself a vector space. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    ⇒   ⊢ (𝜑 → 𝑄 ∈ LVec)
Theorem	algextdeglem4 34394*	Lemma for algextdeg 34399. By lmhmqusker 34010, the surjective module homomorphism 𝐺 described in algextdeglem2 34392 induces an isomorphism with the quotient space. Therefore, the dimension of that quotient space 𝑃 / 𝑍 is the degree of the algebraic field extension. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    ⇒   ⊢ (𝜑 → (dim‘𝑄) = (𝐿[:]𝐾))
Theorem	algextdeglem5 34395*	Lemma for algextdeg 34399. The subspace 𝑍 of annihilators of 𝐴 is a principal ideal generated by the minimal polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    ⇒   ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{(𝑀‘𝐴)}))
Theorem	algextdeglem6 34396*	Lemma for algextdeg 34399. By r1pquslmic 34184, the univariate polynomial remainder ring (𝐻 “s 𝑃) is isomorphic with the quotient ring 𝑄. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    &   ⊢ 𝑅 = (rem1p‘𝐾)    &   ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))    ⇒   ⊢ (𝜑 → (dim‘𝑄) = (dim‘(𝐻 “s 𝑃)))
Theorem	algextdeglem7 34397*	Lemma for algextdeg 34399. The polynomials 𝑋 of lower degree than the minimal polynomial are left unchanged when taking the remainder of the division by that minimal polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    &   ⊢ 𝑅 = (rem1p‘𝐾)    &   ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))    &   ⊢ 𝑇 = (◡(deg1‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝐻‘𝑋) = 𝑋))
Theorem	algextdeglem8 34398*	Lemma for algextdeg 34399. The dimension of the univariate polynomial remainder ring (𝐻 “s 𝑃) is the degree of the minimal polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    &   ⊢ 𝑅 = (rem1p‘𝐾)    &   ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))    &   ⊢ 𝑇 = (◡(deg1‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴))))    ⇒   ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (𝐷‘(𝑀‘𝐴)))
Theorem	algextdeg 34399	The degree of an algebraic field extension (noted [𝐿:𝐾]) is the degree of the minimal polynomial 𝑀(𝐴), whereas 𝐿 is the field generated by 𝐾 and the algebraic element 𝐴. Part of Proposition 1.4 of [Lang], p. 225. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    ⇒   ⊢ (𝜑 → (𝐿[:]𝐾) = (𝐷‘(𝑀‘𝐴)))
21.4.11.4  Quadratic Field Extensions
Theorem	rtelextdg2lem 34400	Lemma for rtelextdg2 34401: If an element 𝑋 is a solution of a quadratic equation, then the degree of its field extension is at most 2. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝑋})))    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑉 = (Base‘𝐸)    &   ⊢ · = (.r‘𝐸)    &   ⊢ + = (+g‘𝐸)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐹)    &   ⊢ (𝜑 → ((2 ↑ 𝑋) + ((𝐴 · 𝑋) + 𝐵)) = 0 )    &   ⊢ 𝑌 = (var1‘𝐾)    &   ⊢ ⊕ = (+g‘𝑃)    &   ⊢ ⊗ = (.r‘𝑃)    &   ⊢ ∧ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑈 = (algSc‘𝑃)    &   ⊢ 𝐺 = ((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))    ⇒   ⊢ (𝜑 → (𝐿[:]𝐾) ≤ 2)