Theorems · Theorem · commutative algebra

integralClosure.mem_lifts_of_monic_of_dvd_map

∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {f : Polynomial R},
  f.Monic →
    ∀ {g : Polynomial K},
      g.Monic → g ∣ Polynomial.map (algebraMap R K) f → g ∈ Polynomial.lifts (algebraMap (↥(integralClosure R K)) K)

Defined in: Mathlib.RingTheory.Polynomial.GaussLemma
Cited by: 1 results in Mathlib
Foundations: Depth 154 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing Field Algebra

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coeproof · cited by 62,936
Setproof · cited by 53,352
CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
RingHomproof · cited by 10,189
SetLike.coeproof · cited by 8,199
Fieldstatement and proof · cited by 7,404
Polynomialstatement and proof · cited by 5,681
Algebra.algebraMapstatement and proof · cited by 4,706
Multisetproof · cited by 2,627
Subalgebrastatement · cited by 1,353
Polynomial.coeffproof · cited by 1,045

Cited by1

Results whose statement or proof uses this declaration.

IsIntegrallyClosed.eq_map_mul_C_of_dvdproof · cited by 2