Theorems · Theorem · field theory
IsPurelyInseparable.surjective_algebraMap_of_isSeparable
∀ (F : Type u_1) (E : Type u_2) [inst : CommRing F] [inst_1 : Ring E] [inst_2 : Algebra F E] [IsPurelyInseparable F E] [Algebra.IsSeparable F E], Function.Surjective ⇑(algebraMap F E)
If E / F is both purely inseparable and separable, then algebraMap F E is surjective.
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 110 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Algebra.algebraMapstatement · cited by 4,706
- Algebra.IsSeparablestatement and proof · cited by 210
- IsPurelyInseparablestatement and proof · cited by 84
- Algebra.IsSeparable.isSeparableproof · cited by 30
- IsPurelyInseparable.inseparableproof · cited by 4
Cited by6
Results whose statement or proof uses this declaration.
- IntermediateField.eq_bot_of_isPurelyInseparable_of_isSeparableproof · cited by 3
- separableClosure.adjoin_eq_of_isAlgebraic_of_isSeparableproof · cited by 2
- IsPurelyInseparable.elemExponent_eq_zero_of_charZeroproof · cited by 1
- IsPurelyInseparable.exists_pow_pow_mem_range_tensorProduct_of_expCharproof · cited by 1
- Subalgebra.eq_bot_of_isPurelyInseparable_of_isSeparableproof · cited by 0
- IsPurelyInseparable.bijective_algebraMap_of_isSeparableproof · cited by 0