  Let $K$ be a local field with finite residue field, we define a normal form
for Eisenstein polynomials depending on the choice of a uniformizer $\pi_K$ and
of residue representatives. The isomorphism classes of extensions generated by
the polynomials in the family exhaust all totally ramified extensions, and the
multiplicity with which each isomorphism class $L/K$ appears is always smaller
than the number of conjugates of $L$ over $K$.
  An algorithm to recover the set of all special polynomials generating the
extension determined by a general Eisenstein polynomial is described. We also
give a criterion to quickly establish that a polynomial generates a different
extension from that generated by a set of special polynomials, such criterion
does not only depend on the usual distance on the set of Eisenstein polynomials
considered by Krasner and others.
  We conclude with an algorithm for the construction of the unique special
equation determining a totally ramified class field in general degree, given a
suitable representation of a group of norms.
