Section AutPolyRoot.
(* The action of automorphisms on roots of unity. *)
Variable F : fieldType.
Implicit Types u v : {rmorphism F -> F}.

(* Specifications: *)
Lemma aut_prim_rootP u z n :
n.-primitive_root z -> {k | coprime k n & u z = z ^+ k}.
Lemma aut_unity_rootP u z n : n > 0 -> z ^+ n = 1 -> {k | u z = z ^+ k}.
Lemma aut_unity_rootC u v z n : n > 0 -> z ^+ n = 1 -> u (v z) = v (u z).
End AutPolyRoot.