Let K be a field and h is an element of K vertical bar z vertical bar be a monic polynomial. By analogy with a relation giving cyclotomic polynomials, for any integer nu >= 1, one defines the nu-th dynatomic polynomial of h to be Phi(nu,h)(z) = Pi(d vertical bar nu)(h(omicron d)(z) - z)(mu(nu/d)). These polynomials are related to the search of primitive periodic points of the dynamic system attached to h. Their studies had been undertaken on one hand by Vivaldi and Hatjispyros [11], on the other hand by Morton and Patel in [5] which give fundamental properties of them. In this paper we are concerned with the polynomials written in the form h(z) = z + g(z), more particularly when g is a distinguished polynomial with coefficients in the valuation ring of a complete ultrametric valued field L with residue characteristic p not equal 0. Let (L) over bar be the residue field of L. For g a distinguished polynomial of degree q a power of p, we obtain in (L) over bar [z] reductions of the polynomials g(nu)(z) = h(omicron nu) (z) - z, nu >= 1 which are additive polynomials with coefficients in (L) over bar independent of g and a reduction of Phi(nu,h), if further nu is a prime number. For a distinguished polynomial of the form g (z) = a(0) + z(p), that is vertical bar a(0)vertical bar < 1, we get that the primitive 3-periodic points of h(z) = a(0) + z + z(p) are the roots of the 3-th dynatomic polynomial Phi(3,h) of h(z). We then study for L equal the field of p-adic numbers the 3-th dynatomic polynomial Phi(3,h) and its roots, when p = 2, 3. For p = 2, we apply Schonemann irreducibility criterion. For p = 5, we use Berlekamp algorithm over the residue field F-5 to establish irreducibility of a polynomial linked to reduction modulo 5 of the 3-th dynatomic polynomial which will be applied elsewhere.
