Section Field.
Variables (F : fieldType) (n : nat) (z : F).
Hypothesis prim_z : n.-primitive_root z.
Let n_gt0 := prim_order_gt0 prim_z.

(* Specifications: *)
Lemma root_cyclotomic x : root (cyclotomic z n) x = n.-primitive_root x.
Lemma prod_cyclotomic :
'X^n - 1 = \prod_(d <- divisors n) cyclotomic (z ^+ (n %/ d)) d.