Abstract : The framework of this thesis is the theory of p-adic representations, in particular Fontaine's theory. I am interested in the case of a metabelian extension of a local field, I build a (phi, Gamma)-module adapted to this extension, then generalizations of some usual tools associated with this (phi, Gamma)-module are given, such as a complex calculating the cohomology of the representation. Furthermore, I establish explicit formulas of the dictionnary between the word of representations and the one of (phi, Gamma)-modules, for the Herr complex, the cup-product or Kummer's map.
The second part of this work is devoted to the proof of Brückner-Vostokov reciprocity law for a formal group. Combining methods of (phi, Gamma)-modules and specified techniques introduced by Abrashkin with a cohomological interpretation of his work, I give a proof of the reciprocity law free from the non natural assumption that roots of unity belong to the base field.