range_eval_eq_rootSet_minpoly — Mathlib

English

If the infinite place v is real, the embedding extension is bijective.

Русский

Если бесконечнаяPlacement v реален, вложение расширения биективно отображается.

LaTeX

$$$\text{IsReal}(v) \Rightarrow \text{Function.Bijective}(\text{extensionEmbeddingOfIsReal}(hv))$$$

Lean4

/-- Let `A` be an algebraically closed field and let `x ∈ K`, with `K` a number field.
The images of `x` by the embeddings of `K` in `A` are exactly the roots in `A` of
the minimal polynomial of `x` over `ℚ`. -/
theorem range_eval_eq_rootSet_minpoly : (range fun φ : K →+* A => φ x) = (minpoly ℚ x).rootSet A :=
  by
  convert (NumberField.isAlgebraic K).range_eval_eq_rootSet_minpoly A x using 1
  ext a
  exact ⟨fun ⟨φ, hφ⟩ => ⟨φ.toRatAlgHom, hφ⟩, fun ⟨φ, hφ⟩ => ⟨φ.toRingHom, hφ⟩⟩

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