Nonempty unramified Extension alg Equiv of is Unramified
IsNonarchimedeanLocalField.nonempty_unramifiedExtension_algEquiv_of_isUnramified
Plain-language statement
If L/K is unramified, then L is isomorphic to Kn where n = [L:K].
Exact Lean statement
theorem nonempty_unramifiedExtension_algEquiv_of_isUnramified [IsUnramified K L] :
Nonempty (UnramifiedExtension K (Module.finrank K L) ≃ₐ[K] L)Formal artifact
Lean source
theorem nonempty_unramifiedExtension_algEquiv_of_isUnramified [IsUnramified K L] : Nonempty (UnramifiedExtension K (Module.finrank K L) ≃ₐ[K] L) := by obtain ⟨φ⟩ := nonempty_unramifiedExtension_alghom_of_dvd_f K L (Module.finrank K L) (IsUnramified.n_dvd_f K L) have : φ.fieldRange = ⊤ := IntermediateField.toSubalgebra_injective <| Subalgebra.toSubmodule_injective <| Submodule.eq_top_of_finrank_eq <| by change Module.finrank K (LinearMap.range φ.toLinearMap) = _ rw [LinearMap.finrank_range_of_inj φ.toRingHom.injective, finrank_unramifiedExtension _ Module.finrank_pos.ne'] exact ⟨(AlgEquiv.ofInjective φ φ.toRingHom.injective).trans <| (IntermediateField.equivOfEq this).trans <| IntermediateField.topEquiv⟩- Project
- Class Field Theory
- License
- Apache-2.0
- Commit
- f18cd7fd1575
- Source
- ClassFieldTheory/IsNonarchimedeanLocalField/Unramified.lean:275-285
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