Theorems · Theorem · field theory
IsGalois.tfae
∀ {F : Type u_1} [inst : Field F] {E : Type u_2} [inst_1 : Field E] [inst_2 : Algebra F E] [FiniteDimensional F E],
[IsGalois F E, IntermediateField.fixedField ⊤ = ⊥, Nat.card Gal(E/F) = Module.finrank F E,
∃ p, p.Separable ∧ Polynomial.IsSplittingField F E p].TFAEEquivalent characterizations of a Galois extension of finite degree.
- Defined in
- Mathlib.FieldTheory.Galois.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Algebrastatement and proof · cited by 11,388
- Top.topstatement and proof · cited by 9,680
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- Bot.botstatement and proof · cited by 4,720
- Subgroupstatement · cited by 3,593
- FiniteDimensionalstatement and proof · cited by 1,854
- Module.finrankstatement and proof · cited by 1,770
- AlgEquivstatement and proof · cited by 1,681
- IntermediateFieldstatement · cited by 988
- Nat.cardstatement and proof · cited by 844
- OrderIso.symmproof · cited by 475
Cited by2
Results whose statement or proof uses this declaration.
- InfiniteGalois.fixedField_fixingSubgroupproof · cited by 3
- Algebra.isInvariant_of_isGaloisproof · cited by 0