Theorems · Theorem · field theory
minpoly.map_eq_of_isSeparable_of_isPurelyInseparable
∀ {F : Type u} (E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {K : Type w} [inst_3 : Field K]
[inst_4 : Algebra F K] [inst_5 : Algebra E K] [IsScalarTower F E K] (x : K),
IsSeparable F x → ∀ [IsPurelyInseparable F E], Polynomial.map (algebraMap F E) (minpoly F x) = minpoly E xIf K / E / F is a field extension tower, such that E / F is purely inseparable, then
for any element x of K separable over F, it has the same minimal polynomials over F and
over E.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 209 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites34
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement · cited by 5,681
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsScalarTowerstatement and proof · cited by 3,896
- Cardinalproof · cited by 2,598
- Module.finrankproof · cited by 1,770
- Polynomial.natDegreeproof · cited by 1,105
- Polynomial.mapstatement · cited by 806
- minpolystatement and proof · cited by 439
- IsIntegralproof · cited by 427
Cited by1
Results whose statement or proof uses this declaration.
- Polynomial.Separable.map_irreducible_of_isPurelyInseparableproof · cited by 0