  We consider a complete discrete valuation field of characteristic p, with
possibly non perfect residue field. Let V be a rank one continuous
representation with finite local monodromy of its absolute Galois group. We
will prove that the Arithmetic Swan conductor of V (defined after Kato in
[Kat89] which fits in the more general theory of [AS02] and [AS06]) coincides
with the Differential Swan conductor of the associated differential module
$D^{\dag}(V)$ defined by Kedlaya in [Ked]. This construction is a
generalization to the non perfect residue case of the Fontaine's formalism as
presented in [Tsu98a]. Our method of proof will allow us to give a new
interpretation of the Refined Swan Conductor.
