Separable.lean`

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1 declarations, 0 contain sorry, 0 declared as axioms

Non-separable polynomials share a monic factor with their derivative

Material for Mathlib.FieldTheory.Separable.

Imports (1)

Mathlib.FieldTheory.Separable

Imported by 2 files

theorem Polynomial.exists_monic_dvd_dvd_derivative_of_not_separable A nonzero polynomial over a field that is not separable has a monic common divisor of positive degree with its derivative, namely its normalized gcd with the derivative line 22 source
Source documentation
A nonzero polynomial over a field that is not separable has a monic common divisor of positive degree with its derivative, namely its normalized gcd with the derivative.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Polynomial, Ne, OfNat.ofNat, Not, Polynomial.Separable, Exists, And, Polynomial.Monic, LT.lt, Nat, Polynomial.natDegree, Dvd.dvd, LinearMap, RingHom.id, Polynomial.derivative
Proof uses: EuclideanDomain.gcd, Classical.propDecidable, Eq, Iff.mp, zero_dvd_iff, Eq.rec, EuclideanDomain.gcd_dvd_left, Exists.intro, normalize, And.intro, Polynomial.monic_normalize, Iff.mpr, Polynomial.Monic.natDegree_pos, Eq.mp, IsCoprime, IsUnit, congrArg, Eq.symm, propext, EuclideanDomain.gcd_isUnit_iff and 4 more
69 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem Polynomial.exists_monic_dvd_dvd_derivative_of_not_separable {k : Type*} [Field k]
    {p : k[X]} (hp : p ≠ 0) (hns : ¬p.Separable) :
    ∃ q : k[X], q.Monic ∧ 0 < q.natDegree ∧ q ∣ p ∧ q ∣ derivative p := by
…
```