splits_of_splits — Mathlib

English

Given f and an embedding i: L → F, the adjoin of the roots in F equals the image i.range.

Русский

Замыкание по корням f в F совпадает с образцом i.

LaTeX

$$$\mathrm{adjoin}_K(\mathrm{rootSet}(f,F)) = i\text{.range}$$$

Lean4

theorem splits_of_splits (h : p.Splits (algebraMap K L)) (hF : ∀ x ∈ p.rootSet L, x ∈ F) : p.Splits (algebraMap K F) :=
  by
  classical
  simp_rw [← F.fieldRange_val, rootSet_def, Finset.mem_coe, Multiset.mem_toFinset] at hF
  exact splits_of_comp _ F.val.toRingHom h hF

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