∀ (R : Type u_1) (k : ℕ) [inst : NeZero k] [inst_1 : CommRing R] [inst_2 : IsDomain R]
  (x : { x // x ∈ Polynomial.nthRoots k 1 }),
  { val := ↑x, inv := ↑x ^ (k - 1), val_inv := ⋯, inv_val := ⋯ } ∈ rootsOfUnity k R