Theorems · Theorem · field theory
Field.exists_primitive_element
- 1000+ list: Primitive element theorem
∀ (F : Type u_1) (E : Type u_2) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] [FiniteDimensional F E] [Algebra.IsSeparable F E], ∃ α, F⟮α⟯ = ⊤
Primitive element theorem: a finite separable field extension E of F has a
primitive element, i.e. there is an α ∈ E such that F⟮α⟯ = (⊤ : Subalgebra F E).
- Defined in
- Mathlib.FieldTheory.PrimitiveElement
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 155 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Algebrastatement and proof · cited by 11,388
- Top.topstatement and proof · cited by 9,680
- Fintypeproof · cited by 7,736
- Fieldstatement and proof · cited by 7,404
- Bot.botproof · cited by 4,720
- FiniteDimensionalstatement and proof · cited by 1,854
- IntermediateFieldstatement and proof · cited by 988
- IsEmptyproof · cited by 759
- IntermediateField.adjoinstatement and proof · cited by 382
- Infiniteproof · cited by 352
- isEmpty_or_nonemptyproof · cited by 269
Cited by7
Results whose statement or proof uses this declaration.
- IsGalois.card_aut_eq_finrankproof · cited by 16
- Field.powerBasisOfFiniteOfSeparableproof · cited by 3
- Algebra.FormallyEtale.of_isSeparableproof · cited by 3
- IsGalois.is_separable_splitting_fieldproof · cited by 2
- Algebra.IsUnramifiedAt.exists_notMem_forall_ne_mem_and_adjoin_eq_topproof · cited by 1
- Differential.differentialAlgebraFiniteDimensionalproof · cited by 0
- IsLocalRing.exists_adjoin_eq_topproof · cited by 0