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feat(RingTheory): generalize relNorm_eq_pow_of_isMaximal to separable fraction field extensions #41696

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feat(RingTheory): generalize relNorm_eq_pow_of_isMaximal to separable fraction field extensions #41696

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2b09020
feat(RingTheory): weaken relNorm_eq_pow_of_isMaximal to separable ext…
vaca22 Jul 13, 2026
068d24d
Remove reference to the previous version from the relNorm_eq_pow_of_i…
vaca22 Jul 18, 2026
b4d76f1
Merge branch 'master' into generalize-relnorm-separable
vaca22 Jul 22, 2026
6c04cca
feat(FieldTheory/Galois): the normal closure of a separable extension…
vaca22 Jul 27, 2026
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19 changes: 18 additions & 1 deletion Mathlib/FieldTheory/Galois/Basic.lean
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Expand Up @@ -575,7 +575,24 @@ end GaloisEquivalentDefinitions
section normalClosure

variable (k K F : Type*) [Field k] [Field K] [Field F] [Algebra k K] [Algebra k F] [Algebra K F]
[IsScalarTower k K F] [IsGalois k F]
[IsScalarTower k K F]

/-- Let $F / K / k$ be a tower of field extensions. If $K$ is separable over $k$,
then the normal closure of $K$ over $k$ in $F$ is separable over $k$. -/
instance IntermediateField.normalClosure.isSeparable [Algebra.IsSeparable k K] :
Algebra.IsSeparable k (normalClosure k K F) := by
rw [← le_separableClosure_iff]
refine normalClosure_le_iff.mpr fun f ↦ ?_
have : Algebra.IsSeparable k f.fieldRange :=
AlgEquiv.Algebra.isSeparable (AlgEquiv.ofInjectiveField f)
exact le_separableClosure k F f.fieldRange
Comment on lines +580 to +588

@xroblot xroblot Jul 31, 2026

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Suggested change
/-- Let $F / K / k$ be a tower of field extensions. If $K$ is separable over $k$,
then the normal closure of $K$ over $k$ in $F$ is separable over $k$. -/
instance IntermediateField.normalClosure.isSeparable [Algebra.IsSeparable k K] :
Algebra.IsSeparable k (normalClosure k K F) := by
rw [← le_separableClosure_iff]
refine normalClosure_le_iff.mpr fun f ↦ ?_
have : Algebra.IsSeparable k f.fieldRange :=
AlgEquiv.Algebra.isSeparable (AlgEquiv.ofInjectiveField f)
exact le_separableClosure k F f.fieldRange
/-- Let `F / K / k` be a tower of field extensions. If `K` is separable over `k`,
then the normal closure of `K/k` in `F` is separable over `k`. -/
instance IntermediateField.normalClosure.isSeparable [Algebra.IsSeparable k K] :
Algebra.IsSeparable k (normalClosure k K F) := by
refine (le_separableClosure_iff _ _ _).mp <| normalClosure_le_iff.mpr fun f ↦ ?_
have : Algebra.IsSeparable k f.fieldRange :=
AlgEquiv.Algebra.isSeparable (AlgEquiv.ofInjectiveField f)
exact le_separableClosure k F f.fieldRange

@xroblot xroblot Jul 31, 2026

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Actually, it seems the $ notation is also used in this file so you can disregard the part about the docstring


/-- Let $F / K / k$ be a tower of field extensions. If $K$ is separable over $k$ and $F$ is normal
over $k$, then the normal closure of $K$ over $k$ in $F$ is Galois over $k$. -/
instance IsGalois.normalClosure_of_isSeparable [Algebra.IsSeparable k K] [Normal k F] :
IsGalois k (normalClosure k K F) where

variable [IsGalois k F]

/-- Let $F / K / k$ be a tower of field extensions. If $F$ is Galois over $k$,
then the normal closure of $K$ over $k$ in $F$ is Galois over $k$. -/
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5 changes: 4 additions & 1 deletion Mathlib/RingTheory/Ideal/Norm/RelNorm.lean
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Expand Up @@ -434,7 +434,10 @@ theorem relNorm_eq_pow_of_isPrime_isGalois [p.IsMaximal] [P.IsPrime]
rw [one_eq_top]
exact IsMaximal.ne_top inferInstance

theorem relNorm_eq_pow_of_isMaximal [PerfectField (FractionRing R)] [P.IsMaximal] [p.IsMaximal] :
/-- The relative norm of a maximal ideal is the prime below it raised to the inertia degree,
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whenever the extension of fraction fields is separable. -/
theorem relNorm_eq_pow_of_isMaximal [Algebra.IsSeparable (FractionRing R) (FractionRing S)]
[P.IsMaximal] [p.IsMaximal] :
relNorm R P = p ^ P.inertiaDeg R := by
let T := Ring.NormalClosure R S
obtain ⟨Q, hQ₁, hQ₂⟩ : ∃ Q : Ideal T, Q.IsMaximal ∧ Q.LiesOver P :=
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2 changes: 1 addition & 1 deletion Mathlib/RingTheory/NormalClosure.lean
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Expand Up @@ -111,7 +111,7 @@ local instance : IsFractionRing T E :=
instance : IsIntegrallyClosed T :=
integralClosure.isIntegrallyClosedOfFiniteExtension L

variable [PerfectField (FractionRing R)]
variable [Algebra.IsSeparable (FractionRing R) (FractionRing S)]

local instance : Algebra.IsSeparable L E :=
Algebra.isSeparable_tower_top_of_isSeparable K L E
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