Intrinsic Approach to Galois Theory of $q$-Difference Equations

The Galois theory of di?erence equations has witnessed a major evolution in the last two decades. In the particular case of q-di?erence equations, authors have introduced several di?erent Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of q-di?erence equations and we use it to establish an arithmetical description of some of the Galois groups attached to q-di?erence systems.