∀ {K : Type u} [inst : Field K] {n : ℕ} {a : K} {L : Type u_1} [inst_1 : Field L] [inst_2 : Algebra K L]
  [inst_3 : Polynomial.IsSplittingField K L (Polynomial.X ^ n - Polynomial.C a)] {α : L}
  (hζ : (primitiveRoots n K).Nonempty) (H : Irreducible (Polynomial.X ^ n - Polynomial.C a))
  (hα : α ^ n = (algebraMap K L) a),
  adjoinRootXPowSubCEquiv hζ H hα =
    AlgEquiv.ofBijective (AdjoinRoot.liftAlgHom (Polynomial.X ^ n - Polynomial.C a) (Algebra.ofId K L) α ⋯) ⋯