Arithmetic equivalence for non-geometric extensions of global function fields

In this paper we study couples of finite separable extensions of the function field Fq(T) which are arithmetically equivalent, i.e. such that prime ideals of Fq[T] decompose with the same inertia degrees in the two fields, up to finitely many exceptions. In the first part of this work, we extend previous results by Cornelissen, Kontogeorgis and Van der Zalm to the case of non-geometric extensions of Fq(T), which are fields such that their field of constants may be bigger than Fq. In the second part, we explicitly produce examples of non-geometric extensions of F2(T) which are equivalent and non-isomorphic over F2(T) and non-equivalent over F4(T), solving a particular Inverse Galois Problem.