Theorem: minpoly.natSepDegree_eq_one_iff_eq_X_sub_C_pow
File: Mathlib/FieldTheory/SeparableDegree.lean

tactic: haveI := expChar_of_injective_algebraMap (algebraMap F E).injective q
state_before: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
⊢ (minpoly F x).natSepDegree = 1 ↔ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
state_after: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this : ExpChar E q
⊢ (minpoly F x).natSepDegree = 1 ↔ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n

tactic: haveI := expChar_of_injective_algebraMap (NoZeroSMulDivisors.algebraMap_injective E E[X]) q
state_before: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this : ExpChar E q
⊢ (minpoly F x).natSepDegree = 1 ↔ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
state_after: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
⊢ (minpoly F x).natSepDegree = 1 ↔ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n

tactic: refine ⟨fun h ↦ ?_, fun ⟨n, h⟩ ↦ (natSepDegree_eq_one_iff_pow_mem q).2 ?_⟩
state_before: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
⊢ (minpoly F x).natSepDegree = 1 ↔ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
state_after: case refine_1
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h : (minpoly F x).natSepDegree = 1
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n

case refine_2
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range

tactic: apply_fun constantCoeff at h
state_before: case refine_2
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
state_after: case refine_2
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : constantCoeff (Polynomial.map (algebraMap F E) (minpoly F x)) = constantCoeff ((X - C x) ^ q ^ n)
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range

tactic: simp_rw [map_pow, map_sub, constantCoeff_apply, coeff_map, coeff_X_zero, coeff_C_zero] at h
state_before: case refine_2
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : constantCoeff (Polynomial.map (algebraMap F E) (minpoly F x)) = constantCoeff ((X - C x) ^ q ^ n)
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
state_after: case refine_2
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : (algebraMap F E) ((minpoly F x).coeff 0) = (0 - x) ^ q ^ n
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range

tactic: rw [zero_sub, neg_pow, ExpChar.neg_one_pow_expChar_pow] at h
state_before: case refine_2
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : (algebraMap F E) ((minpoly F x).coeff 0) = (0 - x) ^ q ^ n
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
state_after: case refine_2
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : (algebraMap F E) ((minpoly F x).coeff 0) = -1 * x ^ q ^ n
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range

tactic: exact ⟨n, -(minpoly F x).coeff 0, by rw [map_neg, h]; ring1⟩
state_before: case refine_2
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : (algebraMap F E) ((minpoly F x).coeff 0) = -1 * x ^ q ^ n
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
state_after: no goals

tactic: obtain ⟨n, y, h⟩ := (natSepDegree_eq_one_iff_eq_X_pow_sub_C q).1 h
state_before: case refine_1
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h : (minpoly F x).natSepDegree = 1
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
state_after: case refine_1.intro.intro
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n

tactic: have hx := congr_arg (Polynomial.aeval x) h.symm
state_before: case refine_1.intro.intro
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
state_after: case refine_1.intro.intro
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
hx : (Polynomial.aeval x) (X ^ q ^ n - C y) = (Polynomial.aeval x) (minpoly F x)
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n

tactic: rw [minpoly.aeval, map_sub, map_pow, aeval_X, aeval_C, sub_eq_zero, eq_comm] at hx
state_before: case refine_1.intro.intro
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
hx : (Polynomial.aeval x) (X ^ q ^ n - C y) = (Polynomial.aeval x) (minpoly F x)
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
state_after: case refine_1.intro.intro
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
hx : (algebraMap F E) y = x ^ q ^ n
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n

tactic: use n
state_before: case refine_1.intro.intro
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
hx : (algebraMap F E) y = x ^ q ^ n
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
state_after: case h
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
hx : (algebraMap F E) y = x ^ q ^ n
⊢ Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n

tactic: rw [h, Polynomial.map_sub, Polynomial.map_pow, map_X, map_C, hx, map_pow, ← sub_pow_expChar_pow]
state_before: case h
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
hx : (algebraMap F E) y = x ^ q ^ n
⊢ Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
state_after: no goals

tactic: rw [map_neg, h]
state_before: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : (algebraMap F E) ((minpoly F x).coeff 0) = -1 * x ^ q ^ n
⊢ (algebraMap F E) (-(minpoly F x).coeff 0) = x ^ q ^ n
state_after: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : (algebraMap F E) ((minpoly F x).coeff 0) = -1 * x ^ q ^ n
⊢ -(-1 * x ^ q ^ n) = x ^ q ^ n

tactic: ring1
state_before: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
x✝ : ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
n : ℕ
h : (algebraMap F E) ((minpoly F x).coeff 0) = -1 * x ^ q ^ n
⊢ -(-1 * x ^ q ^ n) = x ^ q ^ n
state_after: no goals