Theorems · Theorem · field theory
IsGalois.mem_bot_iff_fixed
∀ {F : Type u_1} [inst : Field F] {E : Type u_2} [inst_1 : Field E] [inst_2 : Algebra F E] [IsGalois F E]
[FiniteDimensional F E] (x : E), x ∈ ⊥ ↔ ∀ (f : Gal(E/F)), f x = xSee InfiniteGalois.mem_bot_iff_fixed for the infinite case,
i.e. without the [FiniteDimensional F E] assumption.
- Defined in
- Mathlib.FieldTheory.Galois.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Algebrastatement and proof · cited by 11,388
- Top.topproof · cited by 9,680
- Fieldstatement and proof · cited by 7,404
- Bot.botstatement · cited by 4,720
- FiniteDimensionalstatement and proof · cited by 1,854
- AlgEquivstatement and proof · cited by 1,681
- IntermediateFieldstatement and proof · cited by 988
- IsGaloisstatement and proof · cited by 149
- IntermediateField.mem_fixedField_iffproof · cited by 4
- IsGalois.fixedField_topproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- IsGalois.mem_range_algebraMap_iff_fixedproof · cited by 0
- IntermediateField.restrictRestrictAlgEquivMapHom_surjectiveproof · cited by 0