Theorems · Definition · field theory
Polynomial.IsSplittingField.lift
{F : Type u} →
{K : Type v} →
(L : Type w) →
[inst : Field K] →
[inst_1 : Field L] →
[inst_2 : Field F] →
[inst_3 : Algebra K L] →
[inst_4 : Algebra K F] →
(f : Polynomial K) →
[Polynomial.IsSplittingField K L f] → (Polynomial.map (algebraMap K F) f).Splits → L →ₐ[K] FSplitting field of f embeds into any field that splits f.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 143 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.
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- Algebra.algebraMapstatement and proof · cited by 4,706
- AlgHomstatement · cited by 3,236
- Polynomial.mapstatement and proof · cited by 806
- AlgHom.compproof · cited by 501
- Polynomial.Splitsstatement and proof · cited by 290
- AlgEquiv.toAlgHomproof · cited by 273
- Algebra.ofIdproof · cited by 166
- Polynomial.IsSplittingFieldstatement and proof · cited by 50
- Algebra.botEquivproof · cited by 4
Cited by4
Results whose statement or proof uses this declaration.
- FiniteField.nonempty_algHom_of_finrank_dvdproof · cited by 3
- Polynomial.SplittingField.liftproof · cited by 2
- Polynomial.IsSplittingField.algEquivproof · cited by 1
- Normal.of_isSplittingFieldproof · cited by 0