Theorems · Theorem · commutative algebra
Algebra.IsIntegral.isField_iff_isField
∀ {R : Type u_1} {S : Type u_4} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
[Algebra.IsIntegral R S] [IsDomain S], Function.Injective ⇑(algebraMap R S) → (IsField R ↔ IsField S)- Cited by
- 2 results in Mathlib
- Foundations
- Depth 127 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 and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsDomainstatement and proof · cited by 2,196
- Algebra.IsIntegralstatement and proof · cited by 224
- IsFieldstatement · cited by 103
- isField_of_isIntegral_of_isFieldproof · cited by 4
- isField_of_isIntegral_of_isField'proof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- NumberField.RingOfIntegers.not_isFieldproof · cited by 0
- FunctionField.ringOfIntegers.not_isFieldproof · cited by 0