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.
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