isDomain_of_adjoin_range — Mathlib

English

Given s ⊆ t with hs independence and ht algebraicity, and A domain, there exists a finite transcendence basis between s and t.

Русский

При s ⊆ t с независимостью s и алгебраичностью A над adjoin_R t, существует конечный трансцендентный базис между s и t.

LaTeX

$$$ [NoZeroDivisors A] (s t) (hst : s \\subseteq t) (hs : AlgebraicIndepOn R id s) [ht : Algebra.IsAlgebraic (adjoin R t) A] : \\exists u, s \\subseteq u \\land u \\subseteq t \\land IsTranscendenceBasis R ((\\uparrow) : u \\to A) $$$

Lean4

theorem isDomain_of_adjoin_range [Algebra.IsAlgebraic (adjoin R s) A] : IsDomain A :=
  have := Algebra.IsAlgebraic.nontrivial (adjoin R s) A
  (isDomain_iff_noZeroDivisors_and_nontrivial _).mpr ⟨‹_›, (Subtype.val_injective (p := (· ∈ adjoin R s))).nontrivial⟩

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