isField_of_isAlgClosed_of_isLocalRing

English

If K is algebraically closed, L local ring, then the isomorphism implies IsField L.

Русский

Если K алгебраически замкнуто, L локальное кольцо, тогда изоморфизм имплицирует IsField L.

LaTeX

$$$[IsAlgClosed K] \land [IsLocalRing L] \Rightarrow [IsField L]$$$

Lean4

theorem isField_of_isAlgClosed_of_isLocalRing [IsAlgClosed K] [IsLocalRing A] : IsField A :=
  by
  rw [IsLocalRing.isField_iff_maximalIdeal_eq, eq_bot_iff]
  intro x hx
  obtain ⟨x, rfl⟩ := (bijective_of_isAlgClosed_of_isLocalRing K A).surjective x
  change _ = 0
  rw [← (algebraMap K A).map_zero]
  by_contra hx'
  exact hx ((isUnit_iff_ne_zero.mpr (fun e ↦ hx' ((algebraMap K A).congr_arg e))).map (algebraMap K A))

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