exists_isConj_of_isRamified — Mathlib

English

IsConj φ σ implies IsUnramified k (mk φ).

Русский

IsConj φ σ влечет неразмороженность mk φ.

LaTeX

$$$\\text{IsConj } φ\\; σ \\Rightarrow IsUnramified\\; k\\; (mk φ)$$$

Lean4

theorem exists_isConj_of_isRamified [IsGalois k K] {φ : K →+* ℂ} (h : IsRamified k (mk φ)) :
    ∃ σ : K ≃ₐ[k] K, ComplexEmbedding.IsConj φ σ :=
  by
  rw [isRamified_iff_card_stabilizer_eq_two, Nat.card_eq_two_iff] at h
  obtain ⟨⟨x, hx⟩, ⟨y, hy⟩, h₁, -⟩ := h
  rw [mem_stabilizer_mk_iff] at hx hy
  grind

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