Theorems · Theorem · field theory
Polynomial.Splits.image_rootSet
∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} [inst : CommRing R] [inst_1 : Field A] [inst_2 : Algebra R A]
[inst_3 : CommRing B] [inst_4 : IsDomain B] [inst_5 : Algebra R B] {f : Polynomial R},
(Polynomial.map (algebraMap R A) f).Splits → ∀ (g : A →ₐ[R] B), ⇑g '' f.rootSet A = f.rootSet B- Defined in
- Mathlib.Algebra.Polynomial.Splits
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 141 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites28
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
- Setstatement and proof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Finsetproof · cited by 13,712
- Algebrastatement and proof · cited by 11,388
- RingHomproof · cited by 10,189
- SetLike.coeproof · cited by 8,199
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- Set.imagestatement and proof · cited by 5,609
- Algebra.algebraMapstatement and proof · cited by 4,706
- AlgHomstatement and proof · cited by 3,236
Cited by4
Results whose statement or proof uses this declaration.
- Polynomial.Splits.adjoin_rootSet_eq_rangeproof · cited by 3
- IsIntegral.mem_range_algHom_of_minpoly_splitsproof · cited by 1
- IsGalois.sup_rightproof · cited by 0
- IntermediateField.splits_iff_memproof · cited by 0