Theorems · Theorem · field theory
Polynomial.Splits.roots_map
∀ {R : Type u_1} {S : Type u_2} [inst : Field R] [inst_1 : CommRing S] [inst_2 : IsDomain S] {f : Polynomial R},
f.Splits → ∀ (i : R →+* S), (Polynomial.map i f).roots = Multiset.map (⇑i) f.roots- Defined in
- Mathlib.Algebra.Polynomial.Splits
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 140 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- 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
- Multiset.mapstatement · cited by 876
- Polynomial.mapstatement · cited by 806
- Polynomial.Splitsstatement and proof · cited by 290
- Polynomial.rootsstatement · cited by 264
- RingHom.injectiveproof · cited by 187
Cited by10
Results whose statement or proof uses this declaration.
- Polynomial.Splits.image_rootSetproof · cited by 4
- Polynomial.resultant_eq_prod_evalproof · cited by 3
- Polynomial.natSepDegree_eq_of_splitsproof · cited by 2
- Polynomial.Splits.mem_range_of_isRootproof · cited by 1
- Polynomial.resultant_eq_prod_roots_subproof · cited by 1
- Polynomial.IsSplittingField.mulproof · cited by 0
- Polynomial.Gal.mapRoots_bijectiveproof · cited by 0
- Polynomial.IsSplittingField.splits_iffproof · cited by 0
- Subfield.mem_bot_iff_pow_eq_selfproof · cited by 0
- IsAlgClosed.of_denseRangeproof · cited by 0