mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: R : Type u_4
S : Type u_5
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst✝² : CommRing R
inst✝¹ : Algebra R S
inst✝ : NeZero n
f : S ≃ₐ[R] S
h : ∃ m, f ↑↑hμ.toRootsOfUnity = ↑↑hμ.toRootsOfUnity ^ m
⊢ h.choose % n ≡ h.choose [MOD orderOf hμ.toRootsOfUnity]

Tactic:
]

Goal state after:
R : Type u_4
S : Type u_5
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst✝² : CommRing R
inst✝¹ : Algebra R S
inst✝ : NeZero n
f : S ≃ₐ[R] S
h : ∃ m, f ↑↑hμ.toRootsOfUnity = ↑↑hμ.toRootsOfUnity ^ m
⊢ h.choose % n ≡ h.choose [MOD orderOf hμ.toRootsOfUnity]
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p n : ℕ
hp : Nat.Prime p
m : ℕ
this :
  cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
    (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ i ∈ (p ^ (m + 1)).properDivisors, cyclotomic i R = X ^ p ^ (m + 1) - 1
⊢ cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
    (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
p n : ℕ
hp : Nat.Prime p
m : ℕ
this :
  cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
    (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ i ∈ (p ^ (m + 1)).properDivisors, cyclotomic i R = X ^ p ^ (m + 1) - 1
⊢ cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
    (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
Module: Mathlib.FieldTheory.Galois.Abelian

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Algebra K L
inst✝⁴ : Field M
inst✝³ : Algebra K M
inst✝² : Algebra L M
inst✝¹ : IsScalarTower K L M
inst✝ : IsAbelianGalois K M
this : IsGalois K L
x y : Gal(M/K)
⊢ (AlgEquiv.restrictNormalHom L) (y * x) = (AlgEquiv.restrictNormalHom L) (y * x)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: R : Type u_4
S : Type u_5
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst✝² : CommRing R
inst✝¹ : Algebra R S
inst✝ : NeZero n
f : S ≃ₐ[R] S
h : ∃ m, f ↑↑hμ.toRootsOfUnity = ↑↑hμ.toRootsOfUnity ^ m
⊢ h.choose % orderOf μ ≡ h.choose [MOD orderOf ↑↑hμ.toRootsOfUnity]

Tactic:
]

Goal state after:
R : Type u_4
S : Type u_5
inst✝⁴ : CommRing S
inst✝³ : IsDomain S
μ : S
n : ℕ
hμ : IsPrimitiveRoot μ n
inst✝² : CommRing R
inst✝¹ : Algebra R S
inst✝ : NeZero n
f : S ≃ₐ[R] S
h : ∃ m, f ↑↑hμ.toRootsOfUnity = ↑↑hμ.toRootsOfUnity ^ m
⊢ h.choose % orderOf μ ≡ h.choose [MOD orderOf ↑↑hμ.toRootsOfUnity]
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
inst✝ : NeZero ↑n
hnpos : 0 < n
hμn : orderOf μ ∣ n
hnμ : orderOf μ ≠ n
ho : 0 < orderOf μ
i : ℕ
hio : i ∣ orderOf μ
key : i < n
key' : i ∣ n
hni : {i, n} ⊆ n.divisors
k : K[X]
hk : cyclotomic i K = (X - C μ) * k
j : K[X]
hj : cyclotomic n K = (X - C μ) * j
this : (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) = X ^ n - 1
hn : ∀ (x : K[X]), x * x ∣ (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) → IsUnit x
⊢ False

Tactic:
]

Goal state after:
n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
inst✝ : NeZero ↑n
hnpos : 0 < n
hμn : orderOf μ ∣ n
hnμ : orderOf μ ≠ n
ho : 0 < orderOf μ
i : ℕ
hio : i ∣ orderOf μ
key : i < n
key' : i ∣ n
hni : {i, n} ⊆ n.divisors
k : K[X]
hk : cyclotomic i K = (X - C μ) * k
j : K[X]
hj : cyclotomic n K = (X - C μ) * j
this : (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) = X ^ n - 1
hn : ∀ (x : K[X]), x * x ∣ (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) → IsUnit x
⊢ False
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p n : ℕ
hp : Nat.Prime p
m : ℕ
this :
  cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
    (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ i ∈ (p ^ (m + 1)).properDivisors, cyclotomic i R = X ^ p ^ (m + 1) - 1
⊢ (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1 ↔
    (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
inst✝ : NeZero ↑n
hnpos : 0 < n
hμn : orderOf μ ∣ n
hnμ : orderOf μ ≠ n
ho : 0 < orderOf μ
i : ℕ
hio : i ∣ orderOf μ
key : i < n
key' : i ∣ n
hni : {i, n} ⊆ n.divisors
k : K[X]
hk : cyclotomic i K = (X - C μ) * k
j : K[X]
hj : cyclotomic n K = (X - C μ) * j
this : (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) = X ^ n - 1
hn : ∀ (x : K[X]), x * x ∣ (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) → IsUnit x
⊢ (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) =
    (X - C μ) * (X - C μ) * ((∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * k * j)

Tactic:
by ring

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : ℕ
inst✝¹ : IsDomain R
inst✝ : NeZero ↑n
μ : R
hf : Function.Injective ⇑(algebraMap R (FractionRing R))
this : NeZero ↑n
⊢ (cyclotomic n (FractionRing R)).IsRoot ((algebraMap R (FractionRing R)) μ) ↔
    (cyclotomic n (FractionRing R)).IsRoot ((algebraMap R (FractionRing R)) μ)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: case inr
R : Type u_1
inst✝² : CommRing R
n : ℕ
inst✝¹ : IsDomain R
inst✝ : NeZero ↑n
ζ : R
hζ : IsPrimitiveRoot ζ n
⊢ (cyclotomic n R).roots.Nodup

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝² : CommRing R
n : ℕ
inst✝¹ : IsDomain R
inst✝ : NeZero ↑n
ζ : R
hζ : IsPrimitiveRoot ζ n
⊢ (cyclotomic n R).roots.Nodup
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : ℕ
inst✝¹ : IsDomain R
inst✝ : NeZero ↑n
⊢ (cyclotomic n R).roots = { val := (cyclotomic n R).roots, nodup := ⋯ }.val

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: case inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
m : ℕ
hnm : 1 = cyclotomic m R
⊢ 0 = m

Tactic:
]

Goal state after:
case inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
m : ℕ
hnm : 1 = cyclotomic m R
⊢ 0 = m
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: case inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
m : ℕ
hnm : 0 = m
⊢ 0 = m

Tactic:
]

Goal state after:
case inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
m : ℕ
hnm : 0 = m
⊢ 0 = m
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hnm : map (Int.castRingHom R) (cyclotomic n ℤ) = map (Int.castRingHom R) (cyclotomic m ℤ)
hzero : n ≠ 0
this : NeZero n
⊢ n = m

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hnm : map (Int.castRingHom R) (cyclotomic n ℤ) = map (Int.castRingHom R) (cyclotomic m ℤ)
hzero : n ≠ 0
this : NeZero n
⊢ n = m
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
⊢ n = m

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
⊢ n = m
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n
hroot : (cyclotomic m ℂ).IsRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n))
⊢ n = m

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n
hroot : (cyclotomic m ℂ).IsRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n))
⊢ n = m
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: G : Type u_7
G' : Type u_8
inst✝³ : Group G
inst✝² : IsCyclic G
inst✝¹ : Finite G
inst✝ : CommGroup G'
a : G
ha : a ≠ 1
inst : Fintype G := Fintype.ofFinite G
ζ : G'
hζ : IsPrimitiveRoot ζ (Nat.card G)
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
⊢ Fintype.card G ∣ Fintype.card G

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: case succ
R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
this :
  ∀ (m : ℕ),
    cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
      (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
n_n : ℕ
n_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i ∈ range p, (X ^ p ^ n_n) ^ i
⊢ ?m.258 = ∑ i ∈ range p, (X ^ p ^ (n_n + 1)) ^ i

pp: R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
this :
  ∀ (m : ℕ),
    cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
      (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
n_n : ℕ
n_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i ∈ range p, (X ^ p ^ n_n) ^ i
⊢ R[X]

pp: R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
this :
  ∀ (m : ℕ),
    cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
      (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
n_n : ℕ
n_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i ∈ range p, (X ^ p ^ n_n) ^ i
⊢ ?m.258 * ∏ i ∈ (p ^ (n_n + 1 + 1)).properDivisors, cyclotomic i R = X ^ p ^ (n_n + 1 + 1) - 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
this :
  ∀ (m : ℕ),
    cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
      (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
n_n : ℕ
n_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i ∈ range p, (X ^ p ^ n_n) ^ i
⊢ (∑ i ∈ range p, (X ^ p ^ (n_n + 1)) ^ i) * ∏ i ∈ (p ^ (n_n + 1 + 1)).properDivisors, cyclotomic i R =
    X ^ p ^ (n_n + 1 + 1) - 1
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n
hroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m
hmzero : NeZero m
⊢ n = m

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n
hroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m
hmzero : NeZero m
⊢ n = m
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m
hmzero : NeZero m
hprim : n = m
⊢ n = m

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m
hmzero : NeZero m
hprim : n = m
⊢ n = m
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
this :
  ∀ (m : ℕ),
    cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
      (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
n_n : ℕ
n_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i ∈ range p, (X ^ p ^ n_n) ^ i
⊢ (∑ i ∈ range p, (X ^ p ^ (n_n + 1)) ^ i) *
      ((∏ x ∈ range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i ∈ range p, (X ^ p ^ n_n) ^ i) =
    X ^ p ^ (n_n + 1 + 1) - 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
this :
  ∀ (m : ℕ),
    cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
      (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
n_n : ℕ
n_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i ∈ range p, (X ^ p ^ n_n) ^ i
⊢ (∑ i ∈ range p, (X ^ p ^ (n_n + 1)) ^ i) *
      ((∏ x ∈ range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i ∈ range p, (X ^ p ^ n_n) ^ i) =
    X ^ p ^ (n_n + 1 + 1) - 1
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
this :
  ∀ (m : ℕ),
    cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
      (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
n_n : ℕ
n_ih : (∑ i ∈ range p, (X ^ p ^ n_n) ^ i) * ∏ x ∈ range (n_n + 1), cyclotomic (p ^ x) R = X ^ p ^ (n_n + 1) - 1
⊢ (∑ i ∈ range p, (X ^ p ^ (n_n + 1)) ^ i) *
      ((∏ x ∈ range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i ∈ range p, (X ^ p ^ n_n) ^ i) =
    X ^ p ^ (n_n + 1 + 1) - 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
this :
  ∀ (m : ℕ),
    cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
      (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
n_n : ℕ
n_ih : (∑ i ∈ range p, (X ^ p ^ n_n) ^ i) * ∏ x ∈ range (n_n + 1), cyclotomic (p ^ x) R = X ^ p ^ (n_n + 1) - 1
⊢ (∑ i ∈ range p, (X ^ p ^ (n_n + 1)) ^ i) *
      ((∏ x ∈ range (n_n + 1), cyclotomic (p ^ x) R) * ∑ i ∈ range p, (X ^ p ^ n_n) ^ i) =
    X ^ p ^ (n_n + 1 + 1) - 1
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: G : Type u_7
G' : Type u_8
inst✝³ : Group G
inst✝² : IsCyclic G
inst✝¹ : Finite G
inst✝ : CommGroup G'
a : G
ha : a ≠ 1
inst : Fintype G := Fintype.ofFinite G
ζ : G'
hζ : IsPrimitiveRoot ζ (Nat.card G)
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
hζg : orderOf ζ ∣ orderOf g
φ : G →* G' := monoidHomOfForallMemZpowers hg hζg
⊢ IsPrimitiveRoot ζ (Nat.card G)

Tactic:
]

Goal state after:
G : Type u_7
G' : Type u_8
inst✝³ : Group G
inst✝² : IsCyclic G
inst✝¹ : Finite G
inst✝ : CommGroup G'
a : G
ha : a ≠ 1
inst : Fintype G := Fintype.ofFinite G
ζ : G'
hζ : IsPrimitiveRoot ζ (Nat.card G)
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
hζg : orderOf ζ ∣ orderOf g
φ : G →* G' := monoidHomOfForallMemZpowers hg hζg
⊢ IsPrimitiveRoot ζ (Nat.card G)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: K : Type u_2
inst✝⁴ : Field K
R : Type u_3
inst✝³ : CommRing R
inst✝² : IsDomain R
μ : R
n : ℕ
inst✝¹ : Algebra K R
hμ : IsPrimitiveRoot μ n
h : Irreducible (cyclotomic n K)
inst✝ : NeZero ↑n
this : NeZero ↑n
⊢ IsPrimitiveRoot μ n

Tactic:
]

Goal state after:
K : Type u_2
inst✝⁴ : Field K
R : Type u_3
inst✝³ : CommRing R
inst✝² : IsDomain R
μ : R
n : ℕ
inst✝¹ : Algebra K R
hμ : IsPrimitiveRoot μ n
h : Irreducible (cyclotomic n K)
inst✝ : NeZero ↑n
this : NeZero ↑n
⊢ IsPrimitiveRoot μ n
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
this :
  ∀ (m : ℕ),
    cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
      (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
n_n : ℕ
n_ih : (∑ i ∈ range p, (X ^ p ^ n_n) ^ i) * ∏ x ∈ range (n_n + 1), cyclotomic (p ^ x) R = X ^ p ^ (n_n + 1) - 1
⊢ X ^ (p ^ (n_n + 1) * p) = X ^ p ^ (n_n + 1 + 1)

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
this :
  ∀ (m : ℕ),
    cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
      (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
n_n : ℕ
n_ih : (∑ i ∈ range p, (X ^ p ^ n_n) ^ i) * ∏ x ∈ range (n_n + 1), cyclotomic (p ^ x) R = X ^ p ^ (n_n + 1) - 1
⊢ X ^ (p ^ (n_n + 1) * p) = X ^ p ^ (n_n + 1 + 1)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
h : IsPrimitiveRoot μ n
hpos : 0 < n
inst✝ : CharZero K
⊢ map (Int.castRingHom ℚ) (minpoly ℤ μ) = minpoly ℚ μ

Tactic:
]

Goal state after:
n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
h : IsPrimitiveRoot μ n
hpos : 0 < n
inst✝ : CharZero K
⊢ map (Int.castRingHom ℚ) (minpoly ℤ μ) = minpoly ℚ μ
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: case h
G : Type u_7
G' : Type u_8
inst✝³ : Group G
inst✝² : IsCyclic G
inst✝¹ : Finite G
inst✝ : CommGroup G'
a : G
inst : Fintype G := Fintype.ofFinite G
ζ : G'
hζ : IsPrimitiveRoot ζ (Nat.card G)
g : G
hg✝ : ∀ (x : G), x ∈ Subgroup.zpowers g
hg : ∃ n, g ^ n = a
hζg : orderOf ζ ∣ orderOf g
φ : G →* G' := monoidHomOfForallMemZpowers hg✝ hζg
hφg : IsPrimitiveRoot (φ g) (Nat.card G)
ha : φ a = 1
⊢ a = 1

Tactic:
]

Goal state after:
case h
G : Type u_7
G' : Type u_8
inst✝³ : Group G
inst✝² : IsCyclic G
inst✝¹ : Finite G
inst✝ : CommGroup G'
a : G
inst : Fintype G := Fintype.ofFinite G
ζ : G'
hζ : IsPrimitiveRoot ζ (Nat.card G)
g : G
hg✝ : ∀ (x : G), x ∈ Subgroup.zpowers g
hg : ∃ n, g ^ n = a
hζg : orderOf ζ ∣ orderOf g
φ : G →* G' := monoidHomOfForallMemZpowers hg✝ hζg
hφg : IsPrimitiveRoot (φ g) (Nat.card G)
ha : φ a = 1
⊢ a = 1
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: n : ℕ
hpos : 0 < n
⊢ Irreducible (minpoly ℤ (cexp (2 * ↑Real.pi * I / ↑n)))

Tactic:
]

Goal state after:
n : ℕ
hpos : 0 < n
⊢ Irreducible (minpoly ℤ (cexp (2 * ↑Real.pi * I / ↑n)))
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p k : ℕ
hn : Fact (Nat.Prime p)
⊢ X ^ (p * p ^ k) - 1 = X ^ (p * p ^ k) - 1

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: n : ℕ
hpos : 0 < n
⊢ Irreducible (map (Int.castRingHom ℚ) (cyclotomic n ℤ))

Tactic:
]

Goal state after:
n : ℕ
hpos : 0 < n
⊢ Irreducible (map (Int.castRingHom ℚ) (cyclotomic n ℤ))
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
⊢ (X - 1).coeff 0 * ∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0 = -1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
⊢ (X - 1).coeff 0 * ∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0 = -1
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: case h
G : Type u_7
G' : Type u_8
inst✝³ : Group G
inst✝² : IsCyclic G
inst✝¹ : Finite G
inst✝ : CommGroup G'
a : G
inst : Fintype G := Fintype.ofFinite G
ζ : G'
hζ : IsPrimitiveRoot ζ (Nat.card G)
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
hζg : orderOf ζ ∣ orderOf g
φ : G →* G' := monoidHomOfForallMemZpowers hg hζg
hφg : IsPrimitiveRoot (φ g) (Nat.card G)
k : ℕ
ha : φ g ^ k = 1
hk : g ^ k = a
⊢ a = 1

Tactic:
]

Goal state after:
case h
G : Type u_7
G' : Type u_8
inst✝³ : Group G
inst✝² : IsCyclic G
inst✝¹ : Finite G
inst✝ : CommGroup G'
a : G
inst : Fintype G := Fintype.ofFinite G
ζ : G'
hζ : IsPrimitiveRoot ζ (Nat.card G)
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
hζg : orderOf ζ ∣ orderOf g
φ : G →* G' := monoidHomOfForallMemZpowers hg hζg
hφg : IsPrimitiveRoot (φ g) (Nat.card G)
k : ℕ
ha : φ g ^ k = 1
hk : g ^ k = a
⊢ a = 1
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j
hcongr : ∀ j ∈ n.properDivisors.erase 1, (cyclotomic j R).coeff 0 = 1
⊢ ∏ x ∈ n.properDivisors.erase 1, 1 = 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j
hcongr : ∀ j ∈ n.properDivisors.erase 1, (cyclotomic j R).coeff 0 = 1
⊢ ∏ x ∈ n.properDivisors.erase 1, 1 = 1
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: case inr.inr
n m : ℕ
h : n ≠ m
hnzero : n > 0
hmzero : m > 0
⊢ ¬cyclotomic n ℚ ∣ cyclotomic m ℚ

Tactic:
]

Goal state after:
case inr.inr
n m : ℕ
h : n ≠ m
hnzero : n > 0
hmzero : m > 0
⊢ ¬cyclotomic n ℚ ∣ cyclotomic m ℚ
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1
⊢ -(cyclotomic n R).coeff 0 = -(cyclotomic n R).coeff 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: case h
G : Type u_7
G' : Type u_8
inst✝³ : Group G
inst✝² : IsCyclic G
inst✝¹ : Finite G
inst✝ : CommGroup G'
a : G
inst : Fintype G := Fintype.ofFinite G
ζ : G'
hζ : IsPrimitiveRoot ζ (Nat.card G)
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
hζg : orderOf ζ ∣ orderOf g
φ : G →* G' := monoidHomOfForallMemZpowers hg hζg
hφg : IsPrimitiveRoot (φ g) (Nat.card G)
l : ℕ
ha : φ g ^ (Nat.card G * l) = 1
hk : g ^ (Nat.card G * l) = a
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : CharZero K
p : ℕ
ζ : K
hp : Nat.Prime p
hζ : IsPrimitiveRoot ζ p
α : Fin p → ℚ
this : Fact (Nat.Prime p)
P : ℚ[X] := ∑ i, C (α i) * X ^ ↑i
hP : ∀ (i : Fin p), α i = P.coeff ↑i
x✝¹ : Fin p
x✝ : x✝¹ ∈ Finset.univ
⊢ ↑↑x✝¹ ≤ ↑(p - 1)

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : CharZero K
p : ℕ
ζ : K
hp : Nat.Prime p
hζ : IsPrimitiveRoot ζ p
α : Fin p → ℚ
this : Fact (Nat.Prime p)
P : ℚ[X] := ∑ i, C (α i) * X ^ ↑i
hP : ∀ (i : Fin p), α i = P.coeff ↑i
x✝¹ : Fin p
x✝ : x✝¹ ∈ Finset.univ
⊢ ↑↑x✝¹ ≤ ↑(p - 1)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1
heq : (X ^ n - 1).coeff 0 = -(cyclotomic n R).coeff 0
⊢ eval 0 (X ^ n - 1) = -1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1
heq : (X ^ n - 1).coeff 0 = -(cyclotomic n R).coeff 0
⊢ eval 0 (X ^ n - 1) = -1
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: case h
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1
heq : -1 = -(cyclotomic n R).coeff 0
hzero : (X ^ n - 1).coeff 0 = -1
⊢ (cyclotomic n R).coeff 0 = 1

Tactic:
]

Goal state after:
case h
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1
heq : -1 = -(cyclotomic n R).coeff 0
hzero : (X ^ n - 1).coeff 0 = -1
⊢ (cyclotomic n R).coeff 0 = 1
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: n : ℕ
hpos : 0 < n
p : ℕ
hprime : Fact (Nat.Prime p)
a : ℕ
hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)
⊢ ¬p ∣ a

Tactic:
]

Goal state after:
n : ℕ
hpos : 0 < n
p : ℕ
hprime : Fact (Nat.Prime p)
a : ℕ
hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)
⊢ ¬p ∣ a