P. 341 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34001-34100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fedgmullem2 34001*	Lemma for fedgmul 34002. (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 34002	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 34003	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 34004	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 34005*	In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is a module homomorphism, analogous to ringlghm 20396. (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 34006*	In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 34005. (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 34007	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 34008	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)
21.3.11  Field Extensions
Syntax	cfldext 34009	Syntax for the field extension relation.
class /FldExt
Syntax	cfinext 34010	Syntax for the finite field extension relation.
class /FinExt
Syntax	cextdg 34011	Syntax for the field extension degree operation.
class [:]
Definition	df-fldext 34012*	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 34013*	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 34014*	Definition of the finite field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
Theorem	relfldext 34015	The field extension is a relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ Rel /FldExt
Theorem	brfldext 34016	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 34017	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 34018	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 34019	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 34020	Field extension implies a subring relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ 𝑈 = (Base‘𝐹)    ⇒   ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸))
Theorem	sdrgfldext 34021	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 34022	Field extension implies a structure restriction relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
Theorem	brfinext 34023	The finite field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0))
Theorem	extdgval 34024	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 34025	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 34026	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 34027	Closure of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) ∈ ℕ0*)
Theorem	extdggt0 34028	Degrees of field extension are greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → 0 < (𝐸[:]𝐹))
Theorem	fldexttr 34029	Field extension is a transitive relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
Theorem	fldextid 34030	The field extension relation is reflexive. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ (𝐹 ∈ Field → 𝐹/FldExt𝐹)
Theorem	extdgid 34031	A trivial field extension has degree one. (Contributed by Thierry Arnoux, 4-Aug-2023.)
⊢ (𝐸 ∈ Field → (𝐸[:]𝐸) = 1)
Theorem	fldsdrgfldext 34032	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 34033	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 34034	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 34035	A finite field extension is a field extension. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐸/FinExt𝐹)    ⇒   ⊢ (𝜑 → 𝐸/FldExt𝐹)
Theorem	finexttrb 34036	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 34037	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 34038	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 34039	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 34040	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 34041	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 34042	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 34043	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 34044*	Lemma for fldextrspunlsp 34045: 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 34045	Lemma for fldextrspunfld 34047. 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 34046	Lemma for fldextrspunfld 34047. 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 34047	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 34048	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 34049	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 34050	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 34051	Lemma for fldextrspundgdvds 34052. (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 34052	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 34053*	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))
21.3.11.1  Algebraic numbers
Syntax	cirng 34054	Integral subring of a ring.
class IntgRing
Definition	df-irng 34055*	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 34056*	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 34057*	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 34058	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 34060). (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 34059	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 34060	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 34061	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 34062*	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 34063*	Lemma for extdgfialg 34065. (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 34064*	Lemma for extdgfialg 34065. (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 34065	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)
21.3.11.2  Algebraic extensions
Syntax	calgext 34066	Syntax for the algebraic field extension relation.
class /AlgExt
Definition	df-algext 34067*	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 34068	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 34069	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)
21.3.11.3  Minimal polynomials
Syntax	cminply 34070	Extend class notation with the minimal polynomial builder function.
class minPoly
Definition	df-minply 34071*	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 34072*	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 34073*	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 34074*	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 34075*	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 34076*	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 34075, 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 34077*	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 34078*	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 34079	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 34080	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 34081	Lemma for minplyirred 34082. (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 34082	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 34083	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 34084	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 34085	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 34086	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 34087	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 34088	Lemma for algextdeg 34096. (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 34089*	Lemma for algextdeg 34096. 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 34090*	Lemma for algextdeg 34096. 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 34091*	Lemma for algextdeg 34096. By lmhmqusker 33707, the surjective module homomorphism 𝐺 described in algextdeglem2 34089 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 34092*	Lemma for algextdeg 34096. 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 34093*	Lemma for algextdeg 34096. By r1pquslmic 33881, 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 34094*	Lemma for algextdeg 34096. 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 34095*	Lemma for algextdeg 34096. 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 34096	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)))
21.3.11.4  Quadratic Field Extensions
Theorem	rtelextdg2lem 34097	Lemma for rtelextdg2 34098: 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)
Theorem	rtelextdg2 34098	If an element 𝑋 is a solution of a quadratic equation, then it is either in the base field, or the degree of its field extension is exactly 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 )    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2))
21.3.11.5  Towers of quadratic extentions
Theorem	fldext2chn 34099*	In a non-empty chain 𝑇 of quadratic field extensions, the degree of the final extension is always a power of two. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐸 = (𝑊 ↾s 𝑒)    &   ⊢ 𝐹 = (𝑊 ↾s 𝑓)    &   ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}    &   ⊢ (𝜑 → 𝑇 ∈ ( < Chain (SubDRing‘𝑊)))    &   ⊢ (𝜑 → 𝑊 ∈ Field)    &   ⊢ (𝜑 → (𝑊 ↾s (𝑇‘0)) = 𝑄)    &   ⊢ (𝜑 → (𝑊 ↾s (lastS‘𝑇)) = 𝐿)    &   ⊢ (𝜑 → 0 < (♯‘𝑇))    ⇒   ⊢ (𝜑 → (𝐿/FldExt𝑄 ∧ ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛)))
21.3.12  Constructible Numbers This section defines the set of constructible points as complex numbers which can be drawn starting from two points (we take 0 and 1), and taking intersections of circles and lines. This construction is useful for proving the impossibility of doubling the cube (2sqr3nconstr 34152), and of angle trisection (cos9thpinconstr 34162)
Syntax	cconstr 34100	Extend class notation with the set of constructible points.
class Constr