Theorems · Theorem · field theory
IntermediateField.LinearDisjoint.isField_of_forall
∀ (F : Type u) [inst : Field F] (A : Type v) [inst_1 : Field A] (B : Type w) [inst_2 : Field B] [inst_3 : Algebra F A]
[inst_4 : Algebra F B],
(∀ (K : Type (max v w)) [inst_5 : Field K] [inst_6 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),
fa.fieldRange.LinearDisjoint ↥fb.fieldRange) →
IsField (TensorProduct F A B)If for any field extension K of F that A and B embed into, their images are
linearly disjoint, then A ⊗[F] B is a field. (In the proof we choose K to be the quotient
of A ⊗[F] B by a maximal ideal.)
- Defined in
- Mathlib.FieldTheory.LinearDisjoint
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound
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