  Let $p\neq2$ be a prime. We show a technique based on local class field
theory and on the expansions of certain resultants which allows to recover very
easily Lbekkouri's characterization of Eisenstein polynomials generating cyclic
wild extensions of degree $p^2$ over $\Q_p$, and to extend it to the case of
the base field $K$ being an unramified extension of $\Q_p$.
  Furthermore, when a polynomial satisfies only some of the stated conditions,
we show that the first unsatisfied condition gives information about the Galois
group of the normal closure. This permits to give a complete classification of
Eisenstein polynomials of degree $p^2$ whose splitting field is a
$p$-extension, providing a full description of the Galois group and its higher
ramification subgroups.
  We then apply the same methods to give a characterization of Eisenstein
polynomials of degree $p^3$ generating a cyclic extension.
  In the last section we deduce a combinatorial interpretation of the monomial
symmetric function evaluated in the roots of the unity which appear in certain
expansions.
