mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : IsDomain R
inst✝¹² : Algebra R S
K : Type u_3
L : Type u_4
inst✝¹¹ : Field K
inst✝¹⁰ : Algebra R K
inst✝⁹ : IsFractionRing R K
inst✝⁸ : CommRing L
inst✝⁷ : Nontrivial L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra S L
inst✝⁴ : Algebra K L
inst✝³ : IsScalarTower R K L
inst✝² : IsScalarTower R S L
inst✝¹ : IsIntegrallyClosed R
inst✝ : IsDomain S
s : S
hs : IsIntegral R s
⊢ minpoly K ((algebraMap S L) s) = map (algebraMap R K) (minpoly R s)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.BilinearForm.DualLattice

Goal state before:
pp: R : Type u_4
S : Type u_2
M : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : Field S
inst✝⁵ : AddCommGroup M
inst✝⁴ : Algebra R S
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
B : BilinForm S M
ι : Type u_1
inst✝ : Finite ι
hB : B.Nondegenerate
hB' : B.IsSymm
b : Basis ι S M
⊢ B.dualSubmodule (B.dualSubmodule (Submodule.span R (Set.range ⇑b))) = Submodule.span R (Set.range ⇑b)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : IsDomain R
inst✝⁷ : Algebra R S
K : Type u_3
inst✝⁶ : Field K
inst✝⁵ : Algebra R K
inst✝⁴ : IsFractionRing R K
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : Algebra K S
inst✝ : IsScalarTower R K S
s : S
hs : IsIntegral R s
⊢ minpoly K s = map (algebraMap R K) (minpoly R s)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
⊢ minpolyDiv R x * (X - C x) = map (algebraMap R S) (minpoly R x)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
i : ℕ
⊢ (minpolyDiv R x).coeff i = (algebraMap R S) ((minpoly R x).coeff (i + 1)) + (minpolyDiv R x).coeff (i + 1) * x

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
hx : ¬IsIntegral R x
⊢ minpolyDiv R x = 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
⊢ eval x (minpolyDiv R x) = (aeval x) (derivative (minpoly R x))

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
x : S
inst✝ : IsDomain S
y : S
hxy : y ≠ x
hy : eval y (minpolyDiv R x) = 0 ∨ y - x = 0
⊢ ¬y - x = 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
x : S
inst✝ : IsDomain S
y : S
hxy : y ≠ x
hy : (aeval y) (minpoly R x) = 0
⊢ eval y (minpolyDiv R x) = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
f : R[X]
hf : f.Monic
g : K[X]
hg : g ∣ map (algebraMap R K) f
⊢ ∃ g', map (algebraMap R K) g' * C g.leadingCoeff = g

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
p : R[X]
hp : (Polynomial.aeval s) p = 0
K : Type u_1 := FractionRing R
L : Type u_2 := FractionRing S
x✝ : Algebra K L := FractionRing.liftAlgebra R L
⊢ minpoly K ((algebraMap S L) s) ∣ map (algebraMap R K) (p %ₘ minpoly R s)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
p : R[X]
hp : (Polynomial.aeval s) p = 0
⊢ minpoly R s ∣ p

Tactic:
let K := FractionRing R
  let L := FractionRing S
  let _ : Algebra K L := FractionRing.liftAlgebra R L
  have : minpoly K (algebraMap S L s) ∣ map (algebraMap R K) (p %ₘ minpoly R s) := by
    rw [map_modByMonic _ (minpoly.monic hs), modByMonic_eq_sub_mul_div]
    refine dvd_sub (minpoly.dvd K (algebraMap S L s) ?_) ?_
    · rw [← map_aeval_eq_aeval_map, hp, map_zero]
      rw [← IsScalarTower.algebraMap_eq, ← IsScalarTower.algebraMap_eq]
    apply dvd_mul_of_dvd_left
    rw [isIntegrallyClosed_eq_field_fractions K L hs]
  rw [isIntegrallyClosed_eq_field_fractions _ _ hs,
    map_dvd_map (algebraMap R K) (IsFractionRing.injective R K) (minpoly.monic hs)] at this
  rw [← modByMonic_eq_zero_iff_dvd (minpoly.monic hs)]
  exact Polynomial.eq_zero_of_dvd_of_degree_lt this (degree_modByMonic_lt p <| minpoly.monic hs)

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
p : R[X]
hp : (Polynomial.aeval s) p = 0
⊢ minpoly R s ∣ p

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_3
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
T : Type u_1
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : DecidableEq T
x : S
y : T
σ : S →+* T
hy : eval₂ (σ.comp (algebraMap R S)) y (minpoly R x) = 0
⊢ eval₂ σ y (minpolyDiv R x) = if σ x = y then σ ((aeval x) (derivative (minpoly R x))) else 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
p : R[X]
hp : minpoly R s ∣ p
⊢ (Polynomial.aeval s) p = 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, f x = x
mem : x ∈ (adjoin ↥L {x}).toIntermediateField
this : IntermediateField.fixedField ⊤ = ⊥
⊢ ⟨x, mem⟩ ∈ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
inst✝ : IsDomain S
φ : R →+* S
hinj : Function.Injective ⇑φ
f : R[X]
hf : f.IsPrimitive
⊢ IsUnit f ↔ IsUnit (map φ f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
inst✝ : IsDomain S
φ : R →+* S
hinj : Function.Injective ⇑φ
f : R[X]
hf : f.IsPrimitive
h_irr : Irreducible (map φ f)
a b : R[X]
h : f = a * b
⊢ map φ f = ?m.53 a b h * ?m.54 a b h

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
⊢ RingHom.ker (Polynomial.aeval s).toRingHom = Ideal.span {minpoly R s}

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_1
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : IsDomain T
inst✝ : DecidableEq T
x : S
σ₁ σ₂ : S →ₐ[R] T
⊢ eval₂ (↑σ₁) (σ₂ x) (minpolyDiv R x) = if σ₁ x = σ₂ x then σ₁ ((aeval x) (derivative (minpoly R x))) else 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : IsGalois k K
⊢ IntermediateField.fixedField L.fixingSubgroup = L

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
inst✝ : IsDomain S
φ : R →+* S
hinj : Function.Injective ⇑φ
f : R[X]
hf : f.IsPrimitive
h_irr : Irreducible (map φ f)
⊢ Irreducible f

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
⊢ IntermediateField.fixedField ⊤ = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
p : R[X]
hp0 : p ≠ 0
hp : (Polynomial.aeval s) p = 0
⊢ (minpoly R s).degree ≤ p.degree

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
x : K
⊢ x ∈ ⊥ ↔ ∀ (f : Gal(K/k)), f x = x

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
x : S
inst✝¹ : IsDomain S
inst✝ : DecidableEq S
y : S
hy : (aeval y) (minpoly R x) = 0
⊢ eval y (minpolyDiv R x) = if x = y then (aeval x) (derivative (minpoly R x)) else 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
H : Polynomial.map (algebraMap R K) p = a * b
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
⊢ a.leadingCoeff * b.leadingCoeff = 1

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
x : S
p : R[X]
hirr : Irreducible p
hp : (Polynomial.aeval x) p = 0
int_x : IsIntegral R x
⊢ IsUnit p.leadingCoeff

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
x : S
p : R[X]
hirr : Irreducible p
hp : (Polynomial.aeval x) p = 0
isUnit : IsUnit p.leadingCoeff
⊢ IsIntegral R x

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
i : ℕ
H : (minpolyDiv R x).coeff i ∉ R[x]
⊢ ∀ (j : ℕ), (minpolyDiv R x).coeff (i + j) ∉ R[x]

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
P : R[X]
hmo : P.Monic
hP : (Polynomial.aeval s) P = 0
Pmin : ∀ (Q : R[X]), Q.Monic → (Polynomial.aeval s) Q = 0 → P.degree ≤ Q.degree
hs : IsIntegral R s
hnz : ¬minpoly R s - P = 0
⊢ (Polynomial.aeval s) (minpoly R s - P) = 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
i : ℕ
⊢ (minpolyDiv R x).coeff i ∈ R[x]

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
x : S
hx : IsIntegral R x
inst✝ : Nontrivial S
⊢ minpolyDiv R x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
P : R[X]
hmo : P.Monic
hP : (Polynomial.aeval s) P = 0
Pmin : ∀ (Q : R[X]), Q.Monic → (Polynomial.aeval s) Q = 0 → P.degree ≤ Q.degree
⊢ P = minpoly R s

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
hx : IsIntegral R x
⊢ (minpolyDiv R x).Monic

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ C a.leadingCoeff * C b.leadingCoeff = 1

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
p : R[X]
hp : (Polynomial.aeval s) p = 0
h₀ : p ≠ 0
pmin : ∀ (q : R[X]), q.Monic → (Polynomial.aeval s) q = 0 → p.degree ≤ q.degree
hInt : IsIntegral R s
q : R[X]
hMul : p = minpoly R s * q
this : q.degree ≤ 0
⊢ p = C p.leadingCoeff * minpoly R s

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
x : S
hx : IsIntegral R x
inst✝ : Nontrivial S
⊢ (minpolyDiv R x).natDegree + 1 = (minpoly R x).natDegree

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
x : S
hx : IsIntegral R x
inst✝ : Nontrivial S
⊢ (minpolyDiv R x).natDegree < (minpoly R x).natDegree

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_1
K : Type u_3
S : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
x : S
hx : IsIntegral R x
inst✝⁶ : IsDomain R
inst✝⁵ : IsIntegrallyClosed R
inst✝⁴ : IsDomain S
inst✝³ : Algebra R K
inst✝² : Algebra K S
inst✝¹ : IsScalarTower R K S
inst✝ : IsFractionRing R K
⊢ minpolyDiv R x = minpolyDiv K x

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : Subgroup Gal(K/k)
L : IntermediateField k K
inst✝ : Normal k ↥L
x : K
xL : x ∈ L
h : ∀ a ∈ H, ((restrictNormalHom ↥L) a) ⟨x, xL⟩ = ⟨x, xL⟩
σ : Gal(K/k)
hσ : σ ∈ H
⊢ ↑(((restrictNormalHom ↥L) σ) ⟨x, xL⟩) = x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
h_contra : IsUnit a'
⊢ C a.leadingCoeff * C b.leadingCoeff = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
⊢ Irreducible p ↔ Irreducible (Polynomial.map (algebraMap R K) p)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : Subgroup Gal(K/k)
L : IntermediateField k K
inst✝ : Normal k ↥L
⊢ fixedField H ⊓ L = lift (fixedField (Subgroup.map (restrictNormalHom ↥L) H))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
⊢ IsIntegrallyClosed R ↔ ∀ (p : R[X]), p.Monic → (Irreducible p ↔ Irreducible (map (algebraMap R K) p))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
p q : R[X]
hp : p.Monic
hq : q.Monic
h : Polynomial.map (algebraMap R K) q ∣ Polynomial.map (algebraMap R K) p
⊢ q ∣ p

Tactic:
obtain ⟨r, hr⟩ := h

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
p q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
⊢ q ∣ p
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
p : R[X]
hp : (Polynomial.aeval s) p = 0
h₀ : p ≠ 0
pmin : ∀ (q : R[X]), q.Monic → (Polynomial.aeval s) q = 0 → p.degree ≤ q.degree
x✝ : C p.leadingCoeff ∣ p
q : R[X]
hMul : p = C p.leadingCoeff * q
⊢ p.leadingCoeff ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
p : R[X]
hp : (Polynomial.aeval s) p = 0
h₀ : p ≠ 0
pmin : ∀ (q : R[X]), q.Monic → (Polynomial.aeval s) q = 0 → p.degree ≤ q.degree
⊢ IsIntegral R s ↔ C p.leadingCoeff ∣ p

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
hx : IsIntegral R x
i : ℕ
hi : i ≤ (minpolyDiv R x).natDegree
⊢ (minpolyDiv R x).coeff ((minpolyDiv R x).natDegree - i) - x ^ i ∈ Submodule.span R ((fun x_1 ↦ x ^ x_1) '' Set.Iio i)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
x : S
hx : IsIntegral R x
⊢ Prime (minpoly R x)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : ClosedSubgroup Gal(K/k)
inst✝ : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField ↑H).fixingSubgroup
h : σ ∉ ↑H
b : Set Gal(K/k)
sub : b ⊆ (fun y ↦ σ * y) ⁻¹' (↑H).carrierᶜ
gp : Subgroup Gal(K/k)
L : IntermediateField k K
eq : (fun g ↦ g.carrier) L.fixingSubgroup = b
hL : L ∈ finiteExts k K
eq' : L.fixingSubgroup = gp
this : FiniteDimensional k ↥L
L' : FiniteGaloisIntermediateField k K :=
  let __src := normalClosure k (↥L) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
⊢ σ • L'.fixingSubgroup.carrier ⊆ (↑H).carrierᶜ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
p q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
d' : R[X]
hr' : Polynomial.map (algebraMap R K) d' * C r.leadingCoeff = r
⊢ (Polynomial.map (algebraMap R K) q * r).Monic

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
p q : R[X]
hp : p.Monic
hq : q.Monic
h : Polynomial.map (algebraMap R K) q ∣ Polynomial.map (algebraMap R K) p
⊢ q ∣ p

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
r : R
hr : r ≠ 0
s : S
hs : IsIntegral R s
K : Type u_1 := FractionRing R
L : Type u_2 := FractionRing S
this : Algebra K L := FractionRing.liftAlgebra R L
⊢ (algebraMap R K) (minpoly R s).leadingCoeff ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : K[X]
h0 : p ≠ 0
h : IsUnit (integerNormalization R⁰ p).primPart
u : Rˣ
hu : C ↑u = (integerNormalization R⁰ p).primPart
c : R
c0 : c ∈ R⁰
hc : map (algebraMap R K) (integerNormalization R⁰ p) = C ((algebraMap R K) c) * p
⊢ C ((algebraMap R K) ((integerNormalization R⁰ p).content * ↑u)) =
    map (algebraMap R K) (C ((integerNormalization R⁰ p).content * ↑u))

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
r : R
hr : r ≠ 0
s : S
hs : IsIntegral R s
K : Type u_1 := FractionRing R
L : Type u_2 := FractionRing S
this : Algebra K L := FractionRing.liftAlgebra R L
⊢ ((algebraMap R K) r)⁻¹ ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
r : R
hr : r ≠ 0
s : S
hs : IsIntegral R s
K : Type u_1 := FractionRing R
L : Type u_2 := FractionRing S
this : Algebra K L := FractionRing.liftAlgebra R L
⊢ (algebraMap R K) r ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
r : R
hr : r ≠ 0
s : S
hs : IsIntegral R s
K : Type u_1 := FractionRing R
L : Type u_2 := FractionRing S
this : Algebra K L := FractionRing.liftAlgebra R L
⊢ (algebraMap K L) ((algebraMap R K) r) ≠ 0

Tactic:
by

Goal state after: