Theorems · Theorem · field theory
IsAlgClosed.card_roots_map_eq_natDegree_of_injective
∀ {A : Type u_1} {B : Type u_2} [inst : Semiring A] [inst_1 : Field B] [IsAlgClosed B] {f : A →+* B} (p : Polynomial A),
Function.Injective ⇑f → (Polynomial.map f p).roots.card = p.natDegree- Defined in
- Mathlib.FieldTheory.IsAlgClosed.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 139 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringFieldIsAlgClosed
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.
- DFunLike.coestatement and proof · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- RingHomstatement and proof · cited by 10,189
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- Polynomial.natDegreestatement · cited by 1,105
- Polynomial.mapstatement · cited by 806
- Multiset.cardstatement · cited by 375
- Polynomial.rootsstatement · cited by 264
- IsAlgClosedstatement and proof · cited by 150
- Polynomial.natDegree_map_eq_of_injectiveproof · cited by 11
- IsAlgClosed.card_roots_eq_natDegreeproof · cited by 4
Cited by1
Results whose statement or proof uses this declaration.
- IsAlgClosed.card_aroots_eq_natDegreeproof · cited by 0