-- Mathlib/RingTheory/Trace/Basic.lean
theorem Algebra.isIntegral_trace [FiniteDimensional L F] {x : F} (hx : IsIntegral R x) :
    IsIntegral R (Algebra.trace L F x)

Proof state:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
⊢ Splits (algebraMap L (AlgebraicClosure F)) (minpoly L x)

Tactic:
apply IsAlgClosed.splits_codomain

Result:
no goals

Proof state:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
⊢ IsIntegral R (finrank (↥L⟮x⟯) F • ((minpoly L x).aroots (AlgebraicClosure F)).sum)

Tactic:
refine (IsIntegral.multiset_sum ?_).nsmul _

Result:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
⊢ ∀ x_1 ∈ (minpoly L x).aroots (AlgebraicClosure F), IsIntegral R x_1

Proof state:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
y : AlgebraicClosure F
hy : y ∈ (minpoly L x).aroots (AlgebraicClosure F)
⊢ IsIntegral R y

Tactic:
rw [mem_roots_map (minpoly.ne_zero hx')] at hy

Result:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
y : AlgebraicClosure F
hy : eval₂ (algebraMap L (AlgebraicClosure F)) y (minpoly L x) = 0
⊢ IsIntegral R y

Proof state:
case right
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
y : AlgebraicClosure F
hy : eval₂ (algebraMap L (AlgebraicClosure F)) y (minpoly L x) = 0
⊢ eval₂ (algebraMap R (AlgebraicClosure F)) y (minpoly R x) = 0

Tactic:
rw [← aeval_def] at hy ⊢

Result:
case right
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
y : AlgebraicClosure F
hy : (aeval y) (minpoly L x) = 0
⊢ (aeval y) (minpoly R x) = 0

Proof state:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
⊢ ∀ x_1 ∈ (minpoly L x).aroots (AlgebraicClosure F), IsIntegral R x_1

Tactic:
intro y hy

Result:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
y : AlgebraicClosure F
hy : y ∈ (minpoly L x).aroots (AlgebraicClosure F)
⊢ IsIntegral R y

Proof state:
case right
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
y : AlgebraicClosure F
hy : (aeval y) (minpoly L x) = 0
⊢ (aeval y) (minpoly R x) = 0

Tactic:
exact minpoly.aeval_of_isScalarTower R x y hy

Result:
no goals

Proof state:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
⊢ IsIntegral R ((trace L F) x)

Tactic:
rw [← isIntegral_algebraMap_iff (algebraMap L (AlgebraicClosure F)).injective, trace_eq_sum_roots]

Result:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
⊢ IsIntegral R (finrank (↥L⟮x⟯) F • ((minpoly L x).aroots (AlgebraicClosure F)).sum)

R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
⊢ Splits (algebraMap L (AlgebraicClosure F)) (minpoly L x)

Proof state:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
⊢ IsIntegral R ((trace L F) x)

Tactic:
have hx' : IsIntegral L x := hx.tower_top

Result:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
⊢ IsIntegral R ((trace L F) x)

Proof state:
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
y : AlgebraicClosure F
hy : eval₂ (algebraMap L (AlgebraicClosure F)) y (minpoly L x) = 0
⊢ IsIntegral R y

Tactic:
use minpoly R x, minpoly.monic hx

Result:
case right
R : Type u_1
inst✝⁷ : CommRing R
L : Type u_5
inst✝⁶ : Field L
F : Type u_6
inst✝⁵ : Field F
inst✝⁴ : Algebra R L
inst✝³ : Algebra L F
inst✝² : Algebra R F
inst✝¹ : IsScalarTower R L F
inst✝ : FiniteDimensional L F
x : F
hx : IsIntegral R x
hx' : IsIntegral L x
y : AlgebraicClosure F
hy : eval₂ (algebraMap L (AlgebraicClosure F)) y (minpoly L x) = 0
⊢ eval₂ (algebraMap R (AlgebraicClosure F)) y (minpoly R x) = 0