state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
k : Type u
inst✝ : Field k
H : ∀ (p : k[X]), Monic p → Irreducible p → ∃ x, eval x p = 0
p q : k[X]
hq : Irreducible q
a✝ : q ∣ map (RingHom.id k) p
this : Irreducible (q * C (leadingCoeff q)⁻¹)
⊢ degree q = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | obtain ⟨x, hx⟩ := H (q * C (leadingCoeff q)⁻¹) (monic_mul_leadingCoeff_inv hq.ne_zero) this | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → ∃ x, p.eval x = 0) :
IsAlgClosed k := by
refine ⟨fun p ↦ Or.inr ?_⟩
intro q hq _
have : Irreducible (q * C (leadingCoeff q)⁻¹) := by
rw [← coe_normUnit_of_ne_zero hq.ne_zero]
exact (associated_normalize _).irreducible hq
| Mathlib.FieldTheory.IsAlgClosed.Basic.131_0.PZD1gLxOCbAlYtp | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → ∃ x, p.eval x = 0) :
IsAlgClosed k | Mathlib_FieldTheory_IsAlgClosed_Basic |
case intro
k : Type u
inst✝ : Field k
H : ∀ (p : k[X]), Monic p → Irreducible p → ∃ x, eval x p = 0
p q : k[X]
hq : Irreducible q
a✝ : q ∣ map (RingHom.id k) p
this : Irreducible (q * C (leadingCoeff q)⁻¹)
x : k
hx : eval x (q * C (leadingCoeff q)⁻¹) = 0
⊢ degree q = 1 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | exact degree_mul_leadingCoeff_inv q hq.ne_zero ▸ degree_eq_one_of_irreducible_of_root this hx | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → ∃ x, p.eval x = 0) :
IsAlgClosed k := by
refine ⟨fun p ↦ Or.inr ?_⟩
intro q hq _
have : Irreducible (q * C (leadingCoeff q)⁻¹) := by
rw [← coe_normUnit_of_ne_zero hq.ne_zero]
exact (associated_normalize _).irreducible hq
obtain ⟨x, hx... | Mathlib.FieldTheory.IsAlgClosed.Basic.131_0.PZD1gLxOCbAlYtp | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → ∃ x, p.eval x = 0) :
IsAlgClosed k | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
⊢ IsAlgClosed k' | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | apply IsAlgClosed.of_exists_root | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
| Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case H
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
⊢ ∀ (p : k'[X]), Monic p → Irreducible p → ∃ x, eval x p = 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | intro p hmp hp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
| Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case H
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : Monic p
hp : Irreducible p
⊢ ∃ x, eval x p = 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp) | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
| Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : Monic p
hp : Irreducible p
⊢ degree (map (RingEquiv.toRingHom (RingEquiv.symm e)) p) ≠ 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [degree_map] | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
| Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : Monic p
hp : Irreducible p
⊢ degree p ≠ 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | exact ne_of_gt (degree_pos_of_irreducible hp) | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
| Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case H
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : Monic p
hp : Irreducible p
hpe : degree (map (RingEquiv.toRingHom (RingEquiv.symm e)) p) ≠ 0
⊢ ∃ x, eval x p = 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rcases IsAlgClosed.exists_root (k := k) (p.map e.symm) hpe with ⟨x, hx⟩ | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
| Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case H.intro
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : Monic p
hp : Irreducible p
hpe : degree (map (RingEquiv.toRingHom (RingEquiv.symm e)) p) ≠ 0
x : k
hx : IsRoot (map (↑(RingEquiv.symm e)) p) x
⊢ ∃ x, eval x p = 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | use e x | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := ... | Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : Monic p
hp : Irreducible p
hpe : degree (map (RingEquiv.toRingHom (RingEquiv.symm e)) p) ≠ 0
x : k
hx : IsRoot (map (↑(RingEquiv.symm e)) p) x
⊢ eval (e x) p = 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [IsRoot] at hx | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := ... | Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : Monic p
hp : Irreducible p
hpe : degree (map (RingEquiv.toRingHom (RingEquiv.symm e)) p) ≠ 0
x : k
hx : eval x (map (↑(RingEquiv.symm e)) p) = 0
⊢ eval (e x) p = 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | apply e.symm.injective | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := ... | Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h.a
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : Monic p
hp : Irreducible p
hpe : degree (map (RingEquiv.toRingHom (RingEquiv.symm e)) p) ≠ 0
x : k
hx : eval x (map (↑(RingEquiv.symm e)) p) = 0
⊢ (RingEquiv.symm e) (eval (e x) p) = (RingEquiv.symm e) ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [map_zero, ← hx] | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := ... | Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h.a
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : Monic p
hp : Irreducible p
hpe : degree (map (RingEquiv.toRingHom (RingEquiv.symm e)) p) ≠ 0
x : k
hx : eval x (map (↑(RingEquiv.symm e)) p) = 0
⊢ (RingEquiv.symm e) (eval (e x) p) = eval x (map (↑(Ring... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | clear hx hpe hp hmp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := ... | Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h.a
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
x : k
⊢ (RingEquiv.symm e) (eval (e x) p) = eval x (map (↑(RingEquiv.symm e)) p) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | induction p using Polynomial.induction_on | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := ... | Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h.a.h_C
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
x : k
a✝ : k'
⊢ (RingEquiv.symm e) (eval (e x) (C a✝)) = eval x (map (↑(RingEquiv.symm e)) (C a✝)) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | simp_all | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := ... | Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h.a.h_add
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
x : k
p✝ q✝ : k'[X]
a✝¹ : (RingEquiv.symm e) (eval (e x) p✝) = eval x (map (↑(RingEquiv.symm e)) p✝)
a✝ : (RingEquiv.symm e) (eval (e x) q✝) = eval x (map (↑(RingEquiv.symm e)) q✝)
⊢ (RingEquiv.symm e) (eval (e x... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | simp_all | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := ... | Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h.a.h_monomial
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
x : k
n✝ : ℕ
a✝¹ : k'
a✝ : (RingEquiv.symm e) (eval (e x) (C a✝¹ * X ^ n✝)) = eval x (map (↑(RingEquiv.symm e)) (C a✝¹ * X ^ n✝))
⊢ (RingEquiv.symm e) (eval (e x) (C a✝¹ * X ^ (n✝ + 1))) = eval x (map (↑(Rin... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | simp_all | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := ... | Mathlib.FieldTheory.IsAlgClosed.Basic.142_0.PZD1gLxOCbAlYtp | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' | Mathlib_FieldTheory_IsAlgClosed_Basic |
k✝ : Type u
inst✝⁴ : Field k✝
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Ring K
inst✝¹ : IsDomain K
hk : IsAlgClosed k
inst✝ : Algebra k K
hf : Algebra.IsIntegral k K
⊢ Function.Surjective ⇑(algebraMap k K) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | refine' fun x => ⟨-(minpoly k x).coeff 0, _⟩ | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) := by
| Mathlib.FieldTheory.IsAlgClosed.Basic.162_0.PZD1gLxOCbAlYtp | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsAlgClosed_Basic |
k✝ : Type u
inst✝⁴ : Field k✝
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Ring K
inst✝¹ : IsDomain K
hk : IsAlgClosed k
inst✝ : Algebra k K
hf : Algebra.IsIntegral k K
x : K
⊢ (algebraMap k K) (-coeff (minpoly k x) 0) = x | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (hf x) | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) := by
refine' fun x => ⟨-(minpoly k x).coeff 0, _⟩
| Mathlib.FieldTheory.IsAlgClosed.Basic.162_0.PZD1gLxOCbAlYtp | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsAlgClosed_Basic |
k✝ : Type u
inst✝⁴ : Field k✝
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Ring K
inst✝¹ : IsDomain K
hk : IsAlgClosed k
inst✝ : Algebra k K
hf : Algebra.IsIntegral k K
x : K
hq : leadingCoeff (minpoly k x) = 1
⊢ (algebraMap k K) (-coeff (minpoly k x) 0) = x | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | have h : (minpoly k x).degree = 1 := degree_eq_one_of_irreducible k (minpoly.irreducible (hf x)) | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) := by
refine' fun x => ⟨-(minpoly k x).coeff 0, _⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (hf x... | Mathlib.FieldTheory.IsAlgClosed.Basic.162_0.PZD1gLxOCbAlYtp | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsAlgClosed_Basic |
k✝ : Type u
inst✝⁴ : Field k✝
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Ring K
inst✝¹ : IsDomain K
hk : IsAlgClosed k
inst✝ : Algebra k K
hf : Algebra.IsIntegral k K
x : K
hq : leadingCoeff (minpoly k x) = 1
h : degree (minpoly k x) = 1
⊢ (algebraMap k K) (-coeff (minpoly k x) 0) = x | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | have : aeval x (minpoly k x) = 0 := minpoly.aeval k x | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) := by
refine' fun x => ⟨-(minpoly k x).coeff 0, _⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (hf x... | Mathlib.FieldTheory.IsAlgClosed.Basic.162_0.PZD1gLxOCbAlYtp | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsAlgClosed_Basic |
k✝ : Type u
inst✝⁴ : Field k✝
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Ring K
inst✝¹ : IsDomain K
hk : IsAlgClosed k
inst✝ : Algebra k K
hf : Algebra.IsIntegral k K
x : K
hq : leadingCoeff (minpoly k x) = 1
h : degree (minpoly k x) = 1
this : (aeval x) (minpoly k x) = 0
⊢ (algebraMap k K) (-coeff (minpoly k ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [eq_X_add_C_of_degree_eq_one h, hq, C_1, one_mul, aeval_add, aeval_X, aeval_C,
add_eq_zero_iff_eq_neg] at this | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) := by
refine' fun x => ⟨-(minpoly k x).coeff 0, _⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (hf x... | Mathlib.FieldTheory.IsAlgClosed.Basic.162_0.PZD1gLxOCbAlYtp | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsAlgClosed_Basic |
k✝ : Type u
inst✝⁴ : Field k✝
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Ring K
inst✝¹ : IsDomain K
hk : IsAlgClosed k
inst✝ : Algebra k K
hf : Algebra.IsIntegral k K
x : K
hq : leadingCoeff (minpoly k x) = 1
h : degree (minpoly k x) = 1
this : x = -(algebraMap k K) (coeff (minpoly k x) 0)
⊢ (algebraMap k K) (... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | exact (RingHom.map_neg (algebraMap k K) ((minpoly k x).coeff 0)).symm ▸ this.symm | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) := by
refine' fun x => ⟨-(minpoly k x).coeff 0, _⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (hf x... | Mathlib.FieldTheory.IsAlgClosed.Basic.162_0.PZD1gLxOCbAlYtp | theorem algebraMap_surjective_of_isIntegral {k K : Type*} [Field k] [Ring K] [IsDomain K]
[hk : IsAlgClosed k] [Algebra k K] (hf : Algebra.IsIntegral k K) :
Function.Surjective (algebraMap k K) | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _ | private theorem FractionRing.isAlgebraic :
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI : Algebra (FractionRing R) (FractionRing S) := FractionRing.liftAlgebra R _
Algebra.IsAlgebraic (FractionRing R) (FractionRing S) := by
| Mathlib.FieldTheory.IsAlgClosed.Basic.228_0.PZD1gLxOCbAlYtp | private theorem FractionRing.isAlgebraic :
letI : IsDomain R | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI : Algebra (FractionRing R) (FractionRing S) := FractionRing.liftAlgebra R _ | private theorem FractionRing.isAlgebraic :
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI : Algebra (FractionRing R) (FractionRing S) := FractionRing.liftAlgebra R _
Algebra.IsAlgebraic (FractionRing R) (FractionRing S) := by
letI : IsDomain R := (NoZeroSMulDivisors.alg... | Mathlib.FieldTheory.IsAlgClosed.Basic.228_0.PZD1gLxOCbAlYtp | private theorem FractionRing.isAlgebraic :
letI : IsDomain R | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | have := FractionRing.isScalarTower_liftAlgebra R (FractionRing S) | private theorem FractionRing.isAlgebraic :
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI : Algebra (FractionRing R) (FractionRing S) := FractionRing.liftAlgebra R _
Algebra.IsAlgebraic (FractionRing R) (FractionRing S) := by
letI : IsDomain R := (NoZeroSMulDivisors.alg... | Mathlib.FieldTheory.IsAlgClosed.Basic.228_0.PZD1gLxOCbAlYtp | private theorem FractionRing.isAlgebraic :
letI : IsDomain R | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | intro | private theorem FractionRing.isAlgebraic :
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI : Algebra (FractionRing R) (FractionRing S) := FractionRing.liftAlgebra R _
Algebra.IsAlgebraic (FractionRing R) (FractionRing S) := by
letI : IsDomain R := (NoZeroSMulDivisors.alg... | Mathlib.FieldTheory.IsAlgClosed.Basic.228_0.PZD1gLxOCbAlYtp | private theorem FractionRing.isAlgebraic :
letI : IsDomain R | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | exact
(IsFractionRing.isAlgebraic_iff R (FractionRing R) (FractionRing S)).1
((IsFractionRing.isAlgebraic_iff' R S (FractionRing S)).1 hS _) | private theorem FractionRing.isAlgebraic :
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI : Algebra (FractionRing R) (FractionRing S) := FractionRing.liftAlgebra R _
Algebra.IsAlgebraic (FractionRing R) (FractionRing S) := by
letI : IsDomain R := (NoZeroSMulDivisors.alg... | Mathlib.FieldTheory.IsAlgClosed.Basic.228_0.PZD1gLxOCbAlYtp | private theorem FractionRing.isAlgebraic :
letI : IsDomain R | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _ | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M := by
| Mathlib.FieldTheory.IsAlgClosed.Basic.240_0.PZD1gLxOCbAlYtp | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI := FractionRing.liftAlgebra R M | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M := by
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
| Mathlib.FieldTheory.IsAlgClosed.Basic.240_0.PZD1gLxOCbAlYtp | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI := FractionRing.liftAlgebra R (FractionRing S) | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M := by
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI := FractionRing.liftAlgebra R M
| Mathlib.FieldTheory.IsAlgClosed.Basic.240_0.PZD1gLxOCbAlYtp | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | have := FractionRing.isScalarTower_liftAlgebra R M | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M := by
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI := FractionRing.liftAlgebra R M
letI := FractionRing.liftAlgebra R ... | Mathlib.FieldTheory.IsAlgClosed.Basic.240_0.PZD1gLxOCbAlYtp | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | have := FractionRing.isScalarTower_liftAlgebra R (FractionRing S) | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M := by
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI := FractionRing.liftAlgebra R M
letI := FractionRing.liftAlgebra R ... | Mathlib.FieldTheory.IsAlgClosed.Basic.240_0.PZD1gLxOCbAlYtp | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | have : Algebra.IsAlgebraic (FractionRing R) (FractionRing S) :=
FractionRing.isAlgebraic hS | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M := by
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI := FractionRing.liftAlgebra R M
letI := FractionRing.liftAlgebra R ... | Mathlib.FieldTheory.IsAlgClosed.Basic.240_0.PZD1gLxOCbAlYtp | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | let f : FractionRing S →ₐ[FractionRing R] M := lift_aux (FractionRing R) (FractionRing S) M this | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M := by
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI := FractionRing.liftAlgebra R M
letI := FractionRing.liftAlgebra R ... | Mathlib.FieldTheory.IsAlgClosed.Basic.240_0.PZD1gLxOCbAlYtp | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹³ : Field k
K : Type u
inst✝¹² : Field K
L : Type v
M : Type w
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra K L
inst✝⁹ : Field M
inst✝⁸ : Algebra K M
inst✝⁷ : IsAlgClosed M
hL : Algebra.IsAlgebraic K L
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Algebra R S
inst✝² ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | exact (f.restrictScalars R).comp ((Algebra.ofId S (FractionRing S)).restrictScalars R) | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M := by
letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R S).isDomain _
letI := FractionRing.liftAlgebra R M
letI := FractionRing.liftAlgebra R ... | Mathlib.FieldTheory.IsAlgClosed.Basic.240_0.PZD1gLxOCbAlYtp | /-- A (random) homomorphism from an algebraic extension of R into an algebraically
closed extension of R. -/
noncomputable irreducible_def lift : S →ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algebra R S
... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | apply Infinite.of_not_fintype | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
| Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algeb... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | intro hfin | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
apply Infinite.of_not_fintype
| Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algeb... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | set n := Fintype.card K | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
apply Infinite.of_not_fintype
intro hfin
| Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algeb... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | set f := (X : K[X]) ^ (n + 1) - 1 | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
apply Infinite.of_not_fintype
intro hfin
set n := Fintype.card K
| Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algeb... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | have hfsep : Separable f := separable_X_pow_sub_C 1 (by simp) one_ne_zero | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
apply Infinite.of_not_fintype
intro hfin
set n := Fintype.card K
set f := (X : K[X]) ^ (n + 1) - 1
| Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algebra R S
... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | simp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
apply Infinite.of_not_fintype
intro hfin
set n := Fintype.card K
set f := (X : K[X]) ^ (n + 1) - 1
have hfsep : Separable f := separab... | Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algeb... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | apply Nat.not_succ_le_self (Fintype.card K) | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
apply Infinite.of_not_fintype
intro hfin
set n := Fintype.card K
set f := (X : K[X]) ^ (n + 1) - 1
have hfsep : Separable f := separab... | Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algeb... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | have hroot : n.succ = Fintype.card (f.rootSet K) := by
erw [card_rootSet_eq_natDegree hfsep (IsAlgClosed.splits_domain _), natDegree_X_pow_sub_C] | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
apply Infinite.of_not_fintype
intro hfin
set n := Fintype.card K
set f := (X : K[X]) ^ (n + 1) - 1
have hfsep : Separable f := separab... | Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algebra R S
... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | erw [card_rootSet_eq_natDegree hfsep (IsAlgClosed.splits_domain _), natDegree_X_pow_sub_C] | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
apply Infinite.of_not_fintype
intro hfin
set n := Fintype.card K
set f := (X : K[X]) ^ (n + 1) - 1
have hfsep : Separable f := separab... | Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algeb... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [hroot] | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
apply Infinite.of_not_fintype
intro hfin
set n := Fintype.card K
set f := (X : K[X]) ^ (n + 1) - 1
have hfsep : Separable f := separab... | Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
case h
k : Type u
inst✝¹⁵ : Field k
K✝ : Type u
inst✝¹⁴ : Field K✝
L : Type v
M : Type w
inst✝¹³ : Field L
inst✝¹² : Algebra K✝ L
inst✝¹¹ : Field M
inst✝¹⁰ : Algebra K✝ M
inst✝⁹ : IsAlgClosed M
hL : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝⁸ : CommRing R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algeb... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | exact Fintype.card_le_of_injective _ Subtype.coe_injective | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority := 500) {K : Type*} [Field K] [IsAlgClosed K] : Infinite K := by
apply Infinite.of_not_fintype
intro hfin
set n := Fintype.card K
set f := (X : K[X]) ^ (n + 1) - 1
have hfsep : Separable f := separab... | Mathlib.FieldTheory.IsAlgClosed.Basic.259_0.PZD1gLxOCbAlYtp | /-- Algebraically closed fields are infinite since `Xⁿ⁺¹ - 1` is separable when `#K = n` -/
instance (priority | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝²⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝²³ : Field K
inst✝²² : Field J
inst✝²¹ : CommRing R
inst✝²⁰ : CommRing S
inst✝¹⁹ : Field L
inst✝¹⁸ : Field M
inst✝¹⁷ : Algebra R M
inst✝¹⁶ : NoZeroSMulDivisors R M
inst✝¹⁵ : IsAlgClosure R M
inst✝¹⁴ : Algebra K M
i... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.of_algebraMap_injective <| by
rw [IsScalarTower.algebraMap_eq R S L]
exact (Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective S L)
(NoZeroSMulDivisors.algebraMap_injective R S) : _) | /-- A (random) isomorphism between an algebraic closure of `R` and an algebraic closure of
an algebraic extension of `R` -/
noncomputable def equivOfAlgebraic' [Nontrivial S] [NoZeroSMulDivisors R S]
(hRL : Algebra.IsAlgebraic R L) : L ≃ₐ[R] M := by
| Mathlib.FieldTheory.IsAlgClosed.Basic.315_0.PZD1gLxOCbAlYtp | /-- A (random) isomorphism between an algebraic closure of `R` and an algebraic closure of
an algebraic extension of `R` -/
noncomputable def equivOfAlgebraic' [Nontrivial S] [NoZeroSMulDivisors R S]
(hRL : Algebra.IsAlgebraic R L) : L ≃ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝²⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝²³ : Field K
inst✝²² : Field J
inst✝²¹ : CommRing R
inst✝²⁰ : CommRing S
inst✝¹⁹ : Field L
inst✝¹⁸ : Field M
inst✝¹⁷ : Algebra R M
inst✝¹⁶ : NoZeroSMulDivisors R M
inst✝¹⁵ : IsAlgClosure R M
inst✝¹⁴ : Algebra K M
i... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [IsScalarTower.algebraMap_eq R S L] | /-- A (random) isomorphism between an algebraic closure of `R` and an algebraic closure of
an algebraic extension of `R` -/
noncomputable def equivOfAlgebraic' [Nontrivial S] [NoZeroSMulDivisors R S]
(hRL : Algebra.IsAlgebraic R L) : L ≃ₐ[R] M := by
letI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.of_algebra... | Mathlib.FieldTheory.IsAlgClosed.Basic.315_0.PZD1gLxOCbAlYtp | /-- A (random) isomorphism between an algebraic closure of `R` and an algebraic closure of
an algebraic extension of `R` -/
noncomputable def equivOfAlgebraic' [Nontrivial S] [NoZeroSMulDivisors R S]
(hRL : Algebra.IsAlgebraic R L) : L ≃ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝²⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝²³ : Field K
inst✝²² : Field J
inst✝²¹ : CommRing R
inst✝²⁰ : CommRing S
inst✝¹⁹ : Field L
inst✝¹⁸ : Field M
inst✝¹⁷ : Algebra R M
inst✝¹⁶ : NoZeroSMulDivisors R M
inst✝¹⁵ : IsAlgClosure R M
inst✝¹⁴ : Algebra K M
i... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | exact (Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective S L)
(NoZeroSMulDivisors.algebraMap_injective R S) : _) | /-- A (random) isomorphism between an algebraic closure of `R` and an algebraic closure of
an algebraic extension of `R` -/
noncomputable def equivOfAlgebraic' [Nontrivial S] [NoZeroSMulDivisors R S]
(hRL : Algebra.IsAlgebraic R L) : L ≃ₐ[R] M := by
letI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.of_algebra... | Mathlib.FieldTheory.IsAlgClosed.Basic.315_0.PZD1gLxOCbAlYtp | /-- A (random) isomorphism between an algebraic closure of `R` and an algebraic closure of
an algebraic extension of `R` -/
noncomputable def equivOfAlgebraic' [Nontrivial S] [NoZeroSMulDivisors R S]
(hRL : Algebra.IsAlgebraic R L) : L ≃ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝²⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝²³ : Field K
inst✝²² : Field J
inst✝²¹ : CommRing R
inst✝²⁰ : CommRing S
inst✝¹⁹ : Field L
inst✝¹⁸ : Field M
inst✝¹⁷ : Algebra R M
inst✝¹⁶ : NoZeroSMulDivisors R M
inst✝¹⁵ : IsAlgClosure R M
inst✝¹⁴ : Algebra K M
i... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI : IsAlgClosure R L :=
{ alg_closed := IsAlgClosure.alg_closed S
algebraic := hRL } | /-- A (random) isomorphism between an algebraic closure of `R` and an algebraic closure of
an algebraic extension of `R` -/
noncomputable def equivOfAlgebraic' [Nontrivial S] [NoZeroSMulDivisors R S]
(hRL : Algebra.IsAlgebraic R L) : L ≃ₐ[R] M := by
letI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.of_algebra... | Mathlib.FieldTheory.IsAlgClosed.Basic.315_0.PZD1gLxOCbAlYtp | /-- A (random) isomorphism between an algebraic closure of `R` and an algebraic closure of
an algebraic extension of `R` -/
noncomputable def equivOfAlgebraic' [Nontrivial S] [NoZeroSMulDivisors R S]
(hRL : Algebra.IsAlgebraic R L) : L ≃ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝²⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝²³ : Field K
inst✝²² : Field J
inst✝²¹ : CommRing R
inst✝²⁰ : CommRing S
inst✝¹⁹ : Field L
inst✝¹⁸ : Field M
inst✝¹⁷ : Algebra R M
inst✝¹⁶ : NoZeroSMulDivisors R M
inst✝¹⁵ : IsAlgClosure R M
inst✝¹⁴ : Algebra K M
i... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | exact IsAlgClosure.equiv _ _ _ | /-- A (random) isomorphism between an algebraic closure of `R` and an algebraic closure of
an algebraic extension of `R` -/
noncomputable def equivOfAlgebraic' [Nontrivial S] [NoZeroSMulDivisors R S]
(hRL : Algebra.IsAlgebraic R L) : L ≃ₐ[R] M := by
letI : NoZeroSMulDivisors R L := NoZeroSMulDivisors.of_algebra... | Mathlib.FieldTheory.IsAlgClosed.Basic.315_0.PZD1gLxOCbAlYtp | /-- A (random) isomorphism between an algebraic closure of `R` and an algebraic closure of
an algebraic extension of `R` -/
noncomputable def equivOfAlgebraic' [Nontrivial S] [NoZeroSMulDivisors R S]
(hRL : Algebra.IsAlgebraic R L) : L ≃ₐ[R] M | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
| Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
| Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI : IsDomain R := (NoZeroSMulDivisors.algebraMap_injective R M).isDomain _ | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
| Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI : IsDomain S := (NoZeroSMulDivisors.algebraMap_injective S L).isDomain _ | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | letI : Algebra R L := RingHom.toAlgebra ((algebraMap S L).comp (algebraMap R S)) | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | haveI : IsScalarTower R S L := IsScalarTower.of_algebraMap_eq fun _ => rfl | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | haveI : IsScalarTower S R L :=
IsScalarTower.of_algebraMap_eq (by simp [RingHom.algebraMap_toAlgebra]) | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | simp [RingHom.algebraMap_toAlgebra] | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | haveI : NoZeroSMulDivisors R S := NoZeroSMulDivisors.of_algebraMap_injective hSR.symm.injective | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | refine
⟨equivOfAlgebraic' R S L M
(IsAlgClosure.algebraic.tower_top_of_injective
(show Function.Injective (algebraMap S R) from hSR.injective)), ?_⟩ | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | ext x | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
case a
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | simp only [RingEquiv.toRingHom_eq_coe, Function.comp_apply, RingHom.coe_comp,
AlgEquiv.coe_ringEquiv, RingEquiv.coe_toRingHom] | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
case a
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | conv_lhs => rw [← hSR.symm_apply_apply x] | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [← hSR.symm_apply_apply x] | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [← hSR.symm_apply_apply x] | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [← hSR.symm_apply_apply x] | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
case a
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | show equivOfAlgebraic' R S L M _ (algebraMap R L (hSR x)) = _ | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
case a
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [AlgEquiv.commutes] | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } := by
letI : Algebra R S := RingHom.toAlgebra hSR.symm.toRingHom
letI : Algebra S R := RingHom.toAlgebra hSR.toRingHom
... | Mathlib.FieldTheory.IsAlgClosed.Basic.341_0.PZD1gLxOCbAlYtp | /-- Used in the definition of `equivOfEquiv` -/
noncomputable def equivOfEquivAux (hSR : S ≃+* R) :
{ e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = (algebraMap R M).comp hSR.toRingHom } | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝¹⁴ : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst✝¹³ : Field K
inst✝¹² : Field J
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Field L
inst✝⁸ : Field M
inst✝⁷ : Algebra R M
inst✝⁶ : NoZeroSMulDivisors R M
inst✝⁵ : IsAlgClosure R M
inst✝⁴ : Algebra K M
inst✝³ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | simp | @[simp]
theorem equivOfEquiv_symm_algebraMap (hSR : S ≃+* R) (r : R) :
(equivOfEquiv L M hSR).symm (algebraMap R M r) = algebraMap S L (hSR.symm r) :=
(equivOfEquiv L M hSR).injective (by | Mathlib.FieldTheory.IsAlgClosed.Basic.382_0.PZD1gLxOCbAlYtp | @[simp]
theorem equivOfEquiv_symm_algebraMap (hSR : S ≃+* R) (r : R) :
(equivOfEquiv L M hSR).symm (algebraMap R M r) = algebraMap S L (hSR.symm r) | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝⁵ : Field k
F : Type u_1
K : Type u_2
A : Type u_3
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Field A
inst✝¹ : Algebra F K
inst✝ : Algebra F A
hK : Algebra.IsAlgebraic F K
L : IntermediateField F A
hL : ∀ (x : K), Splits (algebraMap F ↥L) (minpoly F x)
f : K →ₐ[F] A
x : K
⊢ Splits (algebraMap F ↥L) (min... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rw [minpoly.algHom_eq f f.injective] | /-- All `F`-embeddings of a field `K` into another field `A` factor through any intermediate
field of `A/F` in which the minimal polynomial of elements of `K` splits. -/
@[simps]
def IntermediateField.algHomEquivAlgHomOfSplits (L : IntermediateField F A)
(hL : ∀ x : K, (minpoly F x).Splits (algebraMap F L)) :
(... | Mathlib.FieldTheory.IsAlgClosed.Basic.412_0.PZD1gLxOCbAlYtp | /-- All `F`-embeddings of a field `K` into another field `A` factor through any intermediate
field of `A/F` in which the minimal polynomial of elements of `K` splits. -/
@[simps]
def IntermediateField.algHomEquivAlgHomOfSplits (L : IntermediateField F A)
(hL : ∀ x : K, (minpoly F x).Splits (algebraMap F L)) :
(... | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝⁵ : Field k
F : Type u_1
K : Type u_2
A : Type u_3
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Field A
inst✝¹ : Algebra F K
inst✝ : Algebra F A
hK : Algebra.IsAlgebraic F K
L : IntermediateField F A
hL : ∀ (x : K), Splits (algebraMap F ↥L) (minpoly F x)
f : K →ₐ[F] A
x : K
⊢ Splits (algebraMap F ↥L) (min... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | exact hL x | /-- All `F`-embeddings of a field `K` into another field `A` factor through any intermediate
field of `A/F` in which the minimal polynomial of elements of `K` splits. -/
@[simps]
def IntermediateField.algHomEquivAlgHomOfSplits (L : IntermediateField F A)
(hL : ∀ x : K, (minpoly F x).Splits (algebraMap F L)) :
(... | Mathlib.FieldTheory.IsAlgClosed.Basic.412_0.PZD1gLxOCbAlYtp | /-- All `F`-embeddings of a field `K` into another field `A` factor through any intermediate
field of `A/F` in which the minimal polynomial of elements of `K` splits. -/
@[simps]
def IntermediateField.algHomEquivAlgHomOfSplits (L : IntermediateField F A)
(hL : ∀ x : K, (minpoly F x).Splits (algebraMap F L)) :
(... | Mathlib_FieldTheory_IsAlgClosed_Basic |
k : Type u
inst✝⁵ : Field k
F : Type u_1
K : Type u_2
A : Type u_3
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Field A
inst✝¹ : Algebra F K
inst✝ : Algebra F A
hK : Algebra.IsAlgebraic F K
L : IntermediateField F A
hL : ∀ (x : K), Splits (algebraMap F ↥L) (minpoly F x)
x✝ : K →ₐ[F] A
⊢ AlgHom.comp (val L) ((fun f => Alg... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanp... | rfl | /-- All `F`-embeddings of a field `K` into another field `A` factor through any intermediate
field of `A/F` in which the minimal polynomial of elements of `K` splits. -/
@[simps]
def IntermediateField.algHomEquivAlgHomOfSplits (L : IntermediateField F A)
(hL : ∀ x : K, (minpoly F x).Splits (algebraMap F L)) :
(... | Mathlib.FieldTheory.IsAlgClosed.Basic.412_0.PZD1gLxOCbAlYtp | /-- All `F`-embeddings of a field `K` into another field `A` factor through any intermediate
field of `A/F` in which the minimal polynomial of elements of `K` splits. -/
@[simps]
def IntermediateField.algHomEquivAlgHomOfSplits (L : IntermediateField F A)
(hL : ∀ x : K, (minpoly F x).Splits (algebraMap F L)) :
(... | Mathlib_FieldTheory_IsAlgClosed_Basic |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
⊢ Irreducible (factor f) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [factor] | theorem irreducible_factor (f : K[X]) : Irreducible (factor f) := by
| Mathlib.FieldTheory.SplittingField.Construction.54_0.YlgQXigN9SoLViE | theorem irreducible_factor (f : K[X]) : Irreducible (factor f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
⊢ Irreducible (if H : ∃ g, Irreducible g ∧ g ∣ f then Classical.choose H else X) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | split_ifs with H | theorem irreducible_factor (f : K[X]) : Irreducible (factor f) := by
rw [factor]
| Mathlib.FieldTheory.SplittingField.Construction.54_0.YlgQXigN9SoLViE | theorem irreducible_factor (f : K[X]) : Irreducible (factor f) | Mathlib_FieldTheory_SplittingField_Construction |
case pos
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
H : ∃ g, Irreducible g ∧ g ∣ f
⊢ Irreducible (Classical.choose H) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | exact (Classical.choose_spec H).1 | theorem irreducible_factor (f : K[X]) : Irreducible (factor f) := by
rw [factor]
split_ifs with H
· | Mathlib.FieldTheory.SplittingField.Construction.54_0.YlgQXigN9SoLViE | theorem irreducible_factor (f : K[X]) : Irreducible (factor f) | Mathlib_FieldTheory_SplittingField_Construction |
case neg
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
H : ¬∃ g, Irreducible g ∧ g ∣ f
⊢ Irreducible X | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | exact irreducible_X | theorem irreducible_factor (f : K[X]) : Irreducible (factor f) := by
rw [factor]
split_ifs with H
· exact (Classical.choose_spec H).1
· | Mathlib.FieldTheory.SplittingField.Construction.54_0.YlgQXigN9SoLViE | theorem irreducible_factor (f : K[X]) : Irreducible (factor f) | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
hf1 : ¬IsUnit f
⊢ factor f ∣ f | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | by_cases hf2 : f = 0 | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f := by
| Mathlib.FieldTheory.SplittingField.Construction.68_0.YlgQXigN9SoLViE | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f | Mathlib_FieldTheory_SplittingField_Construction |
case pos
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
hf1 : ¬IsUnit f
hf2 : f = 0
⊢ factor f ∣ f | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [hf2] | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f := by
by_cases hf2 : f = 0; · | Mathlib.FieldTheory.SplittingField.Construction.68_0.YlgQXigN9SoLViE | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f | Mathlib_FieldTheory_SplittingField_Construction |
case pos
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
hf1 : ¬IsUnit f
hf2 : f = 0
⊢ factor 0 ∣ 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | exact dvd_zero _ | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f := by
by_cases hf2 : f = 0; · rw [hf2]; | Mathlib.FieldTheory.SplittingField.Construction.68_0.YlgQXigN9SoLViE | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f | Mathlib_FieldTheory_SplittingField_Construction |
case neg
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
hf1 : ¬IsUnit f
hf2 : ¬f = 0
⊢ factor f ∣ f | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)] | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f := by
by_cases hf2 : f = 0; · rw [hf2]; exact dvd_zero _
| Mathlib.FieldTheory.SplittingField.Construction.68_0.YlgQXigN9SoLViE | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f | Mathlib_FieldTheory_SplittingField_Construction |
case neg
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
hf1 : ¬IsUnit f
hf2 : ¬f = 0
⊢ Classical.choose (_ : ∃ i, Irreducible i ∧ i ∣ f) ∣ f | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | exact (Classical.choose_spec <| WfDvdMonoid.exists_irreducible_factor hf1 hf2).2 | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f := by
by_cases hf2 : f = 0; · rw [hf2]; exact dvd_zero _
rw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)]
| Mathlib.FieldTheory.SplittingField.Construction.68_0.YlgQXigN9SoLViE | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
hf : natDegree f ≠ 0
⊢ (X - C (AdjoinRoot.root (factor f))) * removeFactor f = map (AdjoinRoot.of (factor f)) f | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | let ⟨g, hg⟩ := factor_dvd_of_natDegree_ne_zero hf | theorem X_sub_C_mul_removeFactor (f : K[X]) (hf : f.natDegree ≠ 0) :
(X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f := by
| Mathlib.FieldTheory.SplittingField.Construction.87_0.YlgQXigN9SoLViE | theorem X_sub_C_mul_removeFactor (f : K[X]) (hf : f.natDegree ≠ 0) :
(X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
hf : natDegree f ≠ 0
g : K[X]
hg : f = factor f * g
⊢ (X - C (AdjoinRoot.root (factor f))) * removeFactor f = map (AdjoinRoot.of (factor f)) f | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | apply (mul_divByMonic_eq_iff_isRoot
(R := AdjoinRoot f.factor) (a := AdjoinRoot.root f.factor)).mpr | theorem X_sub_C_mul_removeFactor (f : K[X]) (hf : f.natDegree ≠ 0) :
(X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f := by
let ⟨g, hg⟩ := factor_dvd_of_natDegree_ne_zero hf
| Mathlib.FieldTheory.SplittingField.Construction.87_0.YlgQXigN9SoLViE | theorem X_sub_C_mul_removeFactor (f : K[X]) (hf : f.natDegree ≠ 0) :
(X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
hf : natDegree f ≠ 0
g : K[X]
hg : f = factor f * g
⊢ IsRoot (map (AdjoinRoot.of (factor f)) f) (AdjoinRoot.root (factor f)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [IsRoot.def, eval_map, hg, eval₂_mul, ← hg, AdjoinRoot.eval₂_root, zero_mul] | theorem X_sub_C_mul_removeFactor (f : K[X]) (hf : f.natDegree ≠ 0) :
(X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f := by
let ⟨g, hg⟩ := factor_dvd_of_natDegree_ne_zero hf
apply (mul_divByMonic_eq_iff_isRoot
(R := AdjoinRoot f.factor) (a := AdjoinRoot.root f.factor)).mp... | Mathlib.FieldTheory.SplittingField.Construction.87_0.YlgQXigN9SoLViE | theorem X_sub_C_mul_removeFactor (f : K[X]) (hf : f.natDegree ≠ 0) :
(X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
⊢ natDegree (removeFactor f) = natDegree f - 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [removeFactor, natDegree_divByMonic (map (AdjoinRoot.of f.factor) f) (monic_X_sub_C _),
natDegree_map, natDegree_X_sub_C] | theorem natDegree_removeFactor (f : K[X]) : f.removeFactor.natDegree = f.natDegree - 1 := by
-- Porting note: `(map (AdjoinRoot.of f.factor) f)` was `_`
| Mathlib.FieldTheory.SplittingField.Construction.96_0.YlgQXigN9SoLViE | theorem natDegree_removeFactor (f : K[X]) : f.removeFactor.natDegree = f.natDegree - 1 | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]
n : ℕ
hfn : natDegree f = n + 1
⊢ natDegree (removeFactor f) = n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [natDegree_removeFactor, hfn, n.add_sub_cancel] | theorem natDegree_removeFactor' {f : K[X]} {n : ℕ} (hfn : f.natDegree = n + 1) :
f.removeFactor.natDegree = n := by | Mathlib.FieldTheory.SplittingField.Construction.102_0.YlgQXigN9SoLViE | theorem natDegree_removeFactor' {f : K[X]} {n : ℕ} (hfn : f.natDegree = n + 1) :
f.removeFactor.natDegree = n | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n → Splits (algebraMap K (SplittingFieldAux n f)) f)
n
K : Type u
x✝ : Field K
f : K[X]
hf : natDegree f ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [← splits_id_iff_splits, algebraMap_succ, ← map_map, splits_id_iff_splits,
← X_sub_C_mul_removeFactor f fun h => by rw [h] at hf; cases hf] | protected theorem splits (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f :=
Nat.recOn (motive := fun n => ∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f) n
... | Mathlib.FieldTheory.SplittingField.Construction.176_0.YlgQXigN9SoLViE | protected theorem splits (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n → Splits (algebraMap K (SplittingFieldAux n f)) f)
n
K : Type u
x✝ : Field K
f : K[X]
hf : natDegree f ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [h] at hf | protected theorem splits (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f :=
Nat.recOn (motive := fun n => ∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f) n
... | Mathlib.FieldTheory.SplittingField.Construction.176_0.YlgQXigN9SoLViE | protected theorem splits (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n → Splits (algebraMap K (SplittingFieldAux n f)) f)
n
K : Type u
x✝ : Field K
f : K[X]
hf : 0 = Nat.succ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | cases hf | protected theorem splits (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f :=
Nat.recOn (motive := fun n => ∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f) n
... | Mathlib.FieldTheory.SplittingField.Construction.176_0.YlgQXigN9SoLViE | protected theorem splits (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n → Splits (algebraMap K (SplittingFieldAux n f)) f)
n
K : Type u
x✝ : Field K
f : K[X]
hf : natDegree f ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | exact splits_mul _ (splits_X_sub_C _) (ih _ (natDegree_removeFactor' hf)) | protected theorem splits (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f :=
Nat.recOn (motive := fun n => ∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f) n
... | Mathlib.FieldTheory.SplittingField.Construction.176_0.YlgQXigN9SoLViE | protected theorem splits (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | have hndf : f.natDegree ≠ 0 := by intro h; rw [h] at hfn; cases hfn | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | intro h | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [h] at hfn | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | cases hfn | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | have hfn0 : f ≠ 0 := by intro h; rw [h] at hndf; exact hndf rfl | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | intro h | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | rw [h] at hndf | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
F : Type u
K✝¹ : Type v
L : Type w
inst✝³ : Field K✝¹
inst✝² : Field L
inst✝¹ : Field F
n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]),
natDegree f = n → Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = ⊤)
n
K : Type u
x✝ : Field K
f : K[X]
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c49... | exact hndf rfl | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | Mathlib.FieldTheory.SplittingField.Construction.190_0.YlgQXigN9SoLViE | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ | Mathlib_FieldTheory_SplittingField_Construction |
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