It is well known that a finite totally ramified extension of a local field
can be defined by infinitely many Eisenstein polynomials.
Let $ K$ be a finite extension of $Q_{p}$.\\ First O.Ore in [5] and
[6] found some congruencies that must be satisfied by the Eisenstein
polynomials of $K[X]$ of degree $p$ defining cyclic extensions. Then
M.Krasner in [2], with a different target defined an equivalence
relationship between the Eisenstein polynomials defining the same
extension of any degree $n$ over $ K$, then proved the existence of
a privileged representative of each equivalence class which he
called "Reduite". In a previous work in [4, I have considered the
normality problem for an Eisenstein polynomials of degree $p$ and of
degree $p^{2}$ in the case of the base field is $ K=Q_{p}$, when the
residue field is simply $F_{p}$, the finite
field of $p$ elements.\\
The aim of the present article is the explicit determination of such
characteristically polynomials and their Reduites, in the cyclic
case of degree $p$, where the base field is $ K$ a finite extension
of $Q_{p}$. Also illustrating examples are given.