Theorems · Theorem · field theory
Polynomial.map_monic_ne_zero
∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : Semiring S] {f : R →+* S} {p : Polynomial R},
p.Monic → ∀ [Nontrivial S], Polynomial.map f p ≠ 0- Defined in
- Mathlib.Algebra.Polynomial.Eval.Degree
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 108 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringSemiringNontrivial
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- RingHomstatement and proof · cited by 10,189
- Polynomialstatement and proof · cited by 5,681
- Nontrivialstatement and proof · cited by 2,416
- Polynomial.mapstatement and proof · cited by 806
- Polynomial.Monicstatement and proof · cited by 461
- RingHom.map_one_ne_zeroproof · cited by 3
- Polynomial.map_monic_eq_zero_iffproof · cited by 1
Cited by10
Results whose statement or proof uses this declaration.
- IsIntegral.minpoly_splits_tower_top'proof · cited by 3
- PowerSeries.IsWeierstrassFactorizationAt.map_ne_zero_of_ne_topproof · cited by 3
- Ideal.IsFractionRing.normalproof · cited by 3
- Ideal.comap_lt_comap_of_integral_mem_sdiffproof · cited by 2
- IsPrimitiveRoot.is_roots_of_minpolyproof · cited by 1
- IsCyclotomicExtension.Rat.inertiaDeg_eq_of_not_dvdproof · cited by 1
- IsCyclotomicExtension.Rat.ramificationIdx_eq_of_not_dvdproof · cited by 1
- IsAlgClosure.of_splitsproof · cited by 1
- Polynomial.Monic.roots_map_of_card_eq_natDegreeproof · cited by 0