(** if p is prime then sqrt(p) is not rationnal *)
Fact sqrt_prime : forall (p:nat),(is_prime p)->forall (a b:nat),(b<>O)->(p*(square b)<>(square a)).(** We now deduce from this theorem that sqrt(2) is not rationnal *)
(** here is it! *)
Fact sqrt_2_irrat : forall (p q:nat),(q<>O)->(2*(square q)<>(square p)).

Lemma nth_root_irrat_aux : forall (p k a b n r:nat),(is_prime p)->(rel_prime p k)->(0<r)->(r<n)->(rel_prime a b)->((power p r)*k*(power b n) <> (power a n)).