Theorems · Theorem · field theory
IsTranscendenceBasis.isAlgebraic_field
∀ {ι : Type u} {F : Type u_2} {E : Type u_3} {x : ι → E} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E],
IsTranscendenceBasis F x → Algebra.IsAlgebraic (↥(IntermediateField.adjoin F (Set.range x))) E- Cited by
- 3 results in Mathlib
- Foundations
- Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Set.rangestatement and proof · cited by 4,705
- IsScalarTowerproof · cited by 3,896
- IntermediateFieldstatement · cited by 988
- Algebra.adjoinproof · cited by 535
- AlgHom.toRingHomproof · cited by 490
- IntermediateField.adjoinstatement and proof · cited by 382
- RingHom.toAlgebraproof · cited by 337
- Algebra.IsAlgebraicstatement and proof · cited by 322
- IsTranscendenceBasisstatement and proof · cited by 74
Cited by3
Results whose statement or proof uses this declaration.
- finTrdeg_iff_trdegproof · cited by 4
- IsTranscendenceBasis.lift_rank_eq_max_liftproof · cited by 1
- exists_isTranscendenceBasis_and_isSeparable_of_perfectFieldproof · cited by 0