isPurelyInseparable_adjoin_simple_iff_natSepDegree_eq_one

English

A join (sup) of two purely inseparable subextensions over F is still purely inseparable over F.

Русский

Супремум двух чисто неконстантно инseparable-подполей над F остаётся чисто неразделимым над F.

LaTeX

$$IsPurelyInseparable(F, L1 ⊔ L2) from IsPurelyInseparable(F,L1) and IsPurelyInseparable(F,L2)$$

Lean4

/-- `F⟮x⟯ / F` is a purely inseparable extension if and only if the minimal polynomial of `x`
has separable degree one. -/
theorem isPurelyInseparable_adjoin_simple_iff_natSepDegree_eq_one {x : E} :
    IsPurelyInseparable F F⟮x⟯ ↔ (minpoly F x).natSepDegree = 1 := by
  rw [← le_perfectClosure_iff, adjoin_simple_le_iff, mem_perfectClosure_iff_natSepDegree_eq_one]

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