l3lab/ntp-mathlib huggingface dataset:
/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. -/
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) := by
  -- First define some abbreviations.
  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
    hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
  letI := finiteDimensional B
  set P := minpoly R B.gen with hP
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L
  let _ := P.map (algebraMap R L)
  -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
  -- `Q.degree < P.degree` and `Q ≠ 0`.
  rw [adjoin_singleton_eq_range_aeval] at hz
  obtain ⟨Q₁, hQ⟩ := hz
  set Q := Q₁ %ₘ P with hQ₁
  replace hQ : aeval B.gen Q = p • z
  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
    simpa using hQ
  by_cases hQzero : Q = 0
  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
    cases' hQ with H H₁
    · 

Proof state:
case pos.inl
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
hei : IsEisensteinAt P (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ coeff P 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = succ n
this : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQzero : Q = 0
H : (algebraMap R L) p = 0
⊢ z ∈ adjoin R {B.gen}

Next tactic:
have : Function.Injective (algebraMap R L) := by
        rw [algebraMap_eq R K L]
        exact (algebraMap K L).injective.comp (IsFractionRing.injective R K)
/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. -/
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) := by
  -- First define some abbreviations.
  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
    hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
  letI := finiteDimensional B
  set P := minpoly R B.gen with hP
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L
  let _ := P.map (algebraMap R L)
  -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
  -- `Q.degree < P.degree` and `Q ≠ 0`.
  rw [adjoin_singleton_eq_range_aeval] at hz
  obtain ⟨Q₁, hQ⟩ := hz
  set Q := Q₁ %ₘ P with hQ₁
  replace hQ : aeval B.gen Q = p • z
  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
    simpa using hQ
  by_cases hQzero : Q = 0
  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
    cases' hQ with H H₁
    · have : Function.Injective (algebraMap R L) := by
        

Proof state:
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
hei : IsEisensteinAt P (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ coeff P 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = succ n
this : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQzero : Q = 0
H : (algebraMap R L) p = 0
⊢ Function.Injective ⇑(algebraMap R L)

Next tactic:
rw [algebraMap_eq R K L]
/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. -/
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) := by
  -- First define some abbreviations.
  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
    hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
  letI := finiteDimensional B
  set P := minpoly R B.gen with hP
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L
  let _ := P.map (algebraMap R L)
  -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
  -- `Q.degree < P.degree` and `Q ≠ 0`.
  rw [adjoin_singleton_eq_range_aeval] at hz
  obtain ⟨Q₁, hQ⟩ := hz
  set Q := Q₁ %ₘ P with hQ₁
  replace hQ : aeval B.gen Q = p • z
  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
    simpa using hQ
  by_cases hQzero : Q = 0
  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
    cases' hQ with H H₁
    · have : Function.Injective (algebraMap R L) := by
        rw [algebraMap_eq R K L]
        

Proof state:
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
hei : IsEisensteinAt P (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ coeff P 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = succ n
this : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQzero : Q = 0
H : (algebraMap R L) p = 0
⊢ Function.Injective ⇑(RingHom.comp (algebraMap K L) (algebraMap R K))

Next tactic:
exact (algebraMap K L).injective.comp (IsFractionRing.injective R K)
/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. -/
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) := by
  -- First define some abbreviations.
  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
    hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
  letI := finiteDimensional B
  set P := minpoly R B.gen with hP
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L
  let _ := P.map (algebraMap R L)
  -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
  -- `Q.degree < P.degree` and `Q ≠ 0`.
  rw [adjoin_singleton_eq_range_aeval] at hz
  obtain ⟨Q₁, hQ⟩ := hz
  set Q := Q₁ %ₘ P with hQ₁
  replace hQ : aeval B.gen Q = p • z
  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
    simpa using hQ
  by_cases hQzero : Q = 0
  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
    cases' hQ with H H₁
    · have : Function.Injective (algebraMap R L) := by
        rw [algebraMap_eq R K L]
        exact (algebraMap K L).injective.comp (IsFractionRing.injective R K)
      

Proof state:
case pos.inl
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝¹ : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
hei : IsEisensteinAt P (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ coeff P 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = succ n
this✝ : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQzero : Q = 0
H : (algebraMap R L) p = 0
this : Function.Injective ⇑(algebraMap R L)
⊢ z ∈ adjoin R {B.gen}

Next tactic:
exfalso
/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. -/
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) := by
  -- First define some abbreviations.
  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
    hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
  letI := finiteDimensional B
  set P := minpoly R B.gen with hP
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L
  let _ := P.map (algebraMap R L)
  -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
  -- `Q.degree < P.degree` and `Q ≠ 0`.
  rw [adjoin_singleton_eq_range_aeval] at hz
  obtain ⟨Q₁, hQ⟩ := hz
  set Q := Q₁ %ₘ P with hQ₁
  replace hQ : aeval B.gen Q = p • z
  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
    simpa using hQ
  by_cases hQzero : Q = 0
  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
    cases' hQ with H H₁
    · have : Function.Injective (algebraMap R L) := by
        rw [algebraMap_eq R K L]
        exact (algebraMap K L).injective.comp (IsFractionRing.injective R K)
      exfalso
      

Proof state:
case pos.inl.h
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝¹ : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
hei : IsEisensteinAt P (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ coeff P 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = succ n
this✝ : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQzero : Q = 0
H : (algebraMap R L) p = 0
this : Function.Injective ⇑(algebraMap R L)
⊢ False

Next tactic:
exact hp.ne_zero ((injective_iff_map_eq_zero _).1 this _ H)
/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. -/
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) := by
  -- First define some abbreviations.
  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
    hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
  letI := finiteDimensional B
  set P := minpoly R B.gen with hP
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L
  let _ := P.map (algebraMap R L)
  -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
  -- `Q.degree < P.degree` and `Q ≠ 0`.
  rw [adjoin_singleton_eq_range_aeval] at hz
  obtain ⟨Q₁, hQ⟩ := hz
  set Q := Q₁ %ₘ P with hQ₁
  replace hQ : aeval B.gen Q = p • z
  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
    simpa using hQ
  by_cases hQzero : Q = 0
  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
    cases' hQ with H H₁
    · have : Function.Injective (algebraMap R L) := by
        rw [algebraMap_eq R K L]
        exact (algebraMap K L).injective.comp (IsFractionRing.injective R K)
      exfalso
      exact hp.ne_zero ((injective_iff_map_eq_zero _).1 this _ H)
    · 

Proof state:
case pos.inr
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
hei : IsEisensteinAt P (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ coeff P 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = succ n
this : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQzero : Q = 0
H₁ : z = 0
⊢ z ∈ adjoin R {B.gen}

Next tactic:
rw [H₁]
/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. -/
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) := by
  -- First define some abbreviations.
  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
    hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
  letI := finiteDimensional B
  set P := minpoly R B.gen with hP
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L
  let _ := P.map (algebraMap R L)
  -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
  -- `Q.degree < P.degree` and `Q ≠ 0`.
  rw [adjoin_singleton_eq_range_aeval] at hz
  obtain ⟨Q₁, hQ⟩ := hz
  set Q := Q₁ %ₘ P with hQ₁
  replace hQ : aeval B.gen Q = p • z
  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
    simpa using hQ
  by_cases hQzero : Q = 0
  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
    cases' hQ with H H₁
    · have : Function.Injective (algebraMap R L) := by
        rw [algebraMap_eq R K L]
        exact (algebraMap K L).injective.comp (IsFractionRing.injective R K)
      exfalso
      exact hp.ne_zero ((injective_iff_map_eq_zero _).1 this _ H)
    · rw [H₁]
      

Proof state:
case pos.inr
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
hei : IsEisensteinAt P (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ coeff P 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = succ n
this : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQzero : Q = 0
H₁ : z = 0
⊢ 0 ∈ adjoin R {B.gen}

Next tactic:
exact Subalgebra.zero_mem _
/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. -/
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) := by
  -- First define some abbreviations.
  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
    hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
  letI := finiteDimensional B
  set P := minpoly R B.gen with hP
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L
  let _ := P.map (algebraMap R L)
  -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
  -- `Q.degree < P.degree` and `Q ≠ 0`.
  rw [adjoin_singleton_eq_range_aeval] at hz
  obtain ⟨Q₁, hQ⟩ := hz
  set Q := Q₁ %ₘ P with hQ₁
  replace hQ : aeval B.gen Q = p • z
  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
    simpa using hQ
  by_cases hQzero : Q = 0
  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
    cases' hQ with H H₁
    · have : Function.Injective (algebraMap R L) := by
        rw [algebraMap_eq R K L]
        exact (algebraMap K L).injective.comp (IsFractionRing.injective R K)
      exfalso
      exact hp.ne_zero ((injective_iff_map_eq_zero _).1 this _ H)
    · rw [H₁]
      exact Subalgebra.zero_mem _
  -- It is enough to prove that all coefficients of `Q` are divisible by `p`, by induction.
  -- The base case is `dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt`.
  

Proof state:
case neg
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
hei : IsEisensteinAt P (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ coeff P 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = succ n
this : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQ : (aeval B.gen) Q = p • z
hQzero : ¬Q = 0
⊢ z ∈ adjoin R {B.gen}

Next tactic:
refine mem_adjoin_of_dvd_coeff_of_dvd_aeval hp.ne_zero (fun i => ?_) hQ
/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. -/
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) := by
  -- First define some abbreviations.
  have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
    hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
  letI := finiteDimensional B
  set P := minpoly R B.gen with hP
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  haveI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.trans R K L
  let _ := P.map (algebraMap R L)
  -- There is a polynomial `Q` such that `p • z = aeval B.gen Q`. We can assume that
  -- `Q.degree < P.degree` and `Q ≠ 0`.
  rw [adjoin_singleton_eq_range_aeval] at hz
  obtain ⟨Q₁, hQ⟩ := hz
  set Q := Q₁ %ₘ P with hQ₁
  replace hQ : aeval B.gen Q = p • z
  · rw [← modByMonic_add_div Q₁ (minpoly.monic hBint)] at hQ
    simpa using hQ
  by_cases hQzero : Q = 0
  · simp only [hQzero, Algebra.smul_def, zero_eq_mul, aeval_zero] at hQ
    cases' hQ with H H₁
    · have : Function.Injective (algebraMap R L) := by
        rw [algebraMap_eq R K L]
        exact (algebraMap K L).injective.comp (IsFractionRing.injective R K)
      exfalso
      exact hp.ne_zero ((injective_iff_map_eq_zero _).1 this _ H)
    · rw [H₁]
      exact Subalgebra.zero_mem _
  -- It is enough to prove that all coefficients of `Q` are divisible by `p`, by induction.
  -- The base case is `dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt`.
  refine mem_adjoin_of_dvd_coeff_of_dvd_aeval hp.ne_zero (fun i => ?_) hQ
  

Proof state:
case neg
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
hei : IsEisensteinAt P (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ coeff P 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = succ n
this : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQ : (aeval B.gen) Q = p • z
hQzero : ¬Q = 0
i : ℕ
⊢ i ∈ range (natDegree Q + 1) → p ∣ coeff Q i

Next tactic:
induction' i using Nat.case_strong_induction_on with j hind