• Elliptic Curves.
• Global Fields
• Local fields
• Galois Representations

We call a Galois representation a finite dimensional vector space, or a free-module of finite rank over a ring, together with an action of a Galois group associate to some fields extension. Galois representation arise in natural way in algebraic number theory and their studying has a crucial role in this subject. In this thesis we focus our attention to Galois representations that come from elliptic curves, in particular it is based in Serre's Paper "Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques".