Theorems · Theorem · field theory
Polynomial.mem_roots_map
∀ {R : Type u} {k : Type y} [inst : Field R] {p : Polynomial R} [inst_1 : CommRing k] [inst_2 : IsDomain k]
{f : R →+* k} {x : k}, p ≠ 0 → (x ∈ (Polynomial.map f p).roots ↔ Polynomial.eval₂ f x p = 0)- Defined in
- Mathlib.Algebra.Polynomial.FieldDivision
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 134 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- RingHomstatement and proof · cited by 10,189
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- Multisetstatement · cited by 2,627
- IsDomainstatement and proof · cited by 2,196
- Polynomial.mapstatement and proof · cited by 806
- Polynomial.eval₂statement and proof · cited by 267
- Polynomial.rootsstatement · cited by 264
- Polynomial.eval_mapproof · cited by 37
- Polynomial.mem_rootsproof · cited by 30
- Polynomial.map_ne_zeroproof · cited by 22
Cited by2
Results whose statement or proof uses this declaration.
- Algebra.isIntegral_traceproof · cited by 7
- Field.primitive_element_inf_auxproof · cited by 1