Arithmetic equivalence for non-geometric extensions of global function fields

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