Separable.lean`

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1 declarations, 0 contain sorry, 0 declared as axioms

Separability transfer over a discrete valuation ring

Proposed new Mathlib file Mathlib.RingTheory.DiscreteValuationRing.Separable: a monic polynomial over a discrete valuation ring whose reduction is separable stays separable over the fraction field.

Imports (5)

Imported by 2 files

theorem Polynomial.Monic.separable_map_algebraMap_of_separable_map_residue A monic polynomial over a discrete valuation ring R whose reduction to the residue field is separable stays separable over the fraction field K line 43 source
Source documentation
A monic polynomial over a discrete valuation ring R whose reduction to the residue field is separable stays separable over the fraction field K.
Proof: if P.map (algebraMap R K) is not separable, it has a monic common factor g of positive degree with its derivative. Since R is integrally closed and g is a monic divisor of the monic P, Gauss's lemma descends g to a monic Q ∈ R[X], which then divides P and derivative P already in R[X] (map_dvd_map). Reducing mod 𝔪_R, the reduction of Q is a common factor of positive degree of P̄ and derivative P̄, contradicting P̄.Separable.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDomain, IsDiscreteValuationRing, Field, Algebra, IsFractionRing, Polynomial, Polynomial.Monic, Polynomial.Separable, IsLocalRing.ResidueField, Polynomial.map, IsLocalRing.residue, Algebra.algebraMap
Proof uses: Classical.byContradiction, Not, And, LT.lt, Nat, OfNat.ofNat, Polynomial.natDegree, Dvd.dvd, LinearMap, RingHom.id, Polynomial.derivative, Exists, False, Polynomial.exists_monic_dvd_dvd_derivative_of_not_separable, Polynomial.Monic.ne_zero, Polynomial.Monic.map, Eq, Polynomial.Monic.exists_monic_map_eq_of_monic_dvd_map, Function.Injective, RingHom and 14 more
47 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
theorem Polynomial.Monic.separable_map_algebraMap_of_separable_map_residue {P : R[X]}
    (hmonic : P.Monic) (hsep : (P.map (residue R)).Separable) :
    (P.map (algebraMap R K)).Separable := by
…
```