GaussLemma.lean`

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1 declarations, 0 contain sorry, 0 declared as axioms

Monic divisors descend along the fraction field

Material for Mathlib.RingTheory.Polynomial.GaussLemma.

Imports (1)

Mathlib.RingTheory.Polynomial.GaussLemma

Imported by 2 files

theorem Polynomial.Monic.exists_monic_map_eq_of_monic_dvd_map Documented declaration: exists_monic_map_eq_of_monic_dvd_map line 23 source
Source documentation
Over an integrally closed domain R with fraction field K, a monic divisor q ∈ K[X] of the image of a monic polynomial P ∈ R[X] is itself the image of a monic polynomial over R (a form of Gauss's lemma, packaging IsIntegrallyClosed.eq_map_mul_C_of_dvd).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsIntegrallyClosed, Field, Algebra, IsFractionRing, Polynomial, Polynomial.Monic, Dvd.dvd, Polynomial.map, Algebra.algebraMap, Exists, And, Eq
Proof uses: HMul.hMul, RingHom, Polynomial.C, Polynomial.leadingCoeff, IsIntegrallyClosed.eq_map_mul_C_of_dvd, Exists.intro, And.intro, Iff.mpr, Function.Injective.monic_map_iff, IsFractionRing.injective, Eq.rec, Eq.symm, Eq.mp, OfNat.ofNat, congrArg, mul_one, Polynomial.C_1, Polynomial.Monic.leadingCoeff
41 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem Polynomial.Monic.exists_monic_map_eq_of_monic_dvd_map {R : Type*} [CommRing R]
    [IsIntegrallyClosed R] {K : Type*} [Field K] [Algebra R K] [IsFractionRing R K]
    {P : R[X]} (hP : P.Monic) {q : K[X]} (hq : q.Monic) (hdvd : q ∣ P.map (algebraMap R K)) :
    ∃ Q : R[X], Q.Monic ∧ Q.map (algebraMap R K) = q := by
…
```