Theorems · Theorem · commutative algebra

Polynomial.IsWeaklyEisensteinAt.pow_natDegree_le_of_aeval_zero_of_monic_mem_map

∀ {R : Type u} [inst : CommRing R] {𝓟 : Ideal R} {f : Polynomial R} {S : Type v} [inst_1 : CommRing S]
  [inst_2 : Algebra R S],
  f.IsWeaklyEisensteinAt 𝓟 →
    ∀ {x : S},
      (Polynomial.aeval x) f = 0 →
        f.Monic → ∀ (i : ℕ), (Polynomial.map (algebraMap R S) f).natDegree ≤ i → x ^ i ∈ Ideal.map (algebraMap R S) 𝓟

Defined in: Mathlib.RingTheory.Polynomial.Eisenstein.Basic
Cited by: 0 results in Mathlib
Foundations: Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing CommRing Algebra

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DFunLike.coestatement and proof · cited by 62,936
CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
RingHomstatement · cited by 10,189
Polynomialstatement and proof · cited by 5,681
Idealstatement and proof · cited by 4,748
Algebra.algebraMapstatement and proof · cited by 4,706
AlgHomstatement · cited by 3,236
Polynomial.natDegreestatement and proof · cited by 1,105
Polynomial.mapstatement and proof · cited by 806
Ideal.mapstatement and proof · cited by 692
Polynomial.aevalstatement and proof · cited by 615

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