The main purpose of this paper is to describe the abelian part G ab K of the absolute Galois group of a global function field K as pro-finite group. We will show that the characteristic p of K and the non p-part of the class group of K are determined by G ab K . The converse is almost true: isomorphism type of G ab K as profinite group is determined by the invariant d K of the constant field F q introduced in first section and the non p-part of the class group.
In this short note we provide a few examples of non-isomorphic arithmetically equivalent global function fields. These examples are obtained via well-known technique of adjoining the torsion points of various Drinfeld Modules to realise the Gl n (F q ) as a Galois group of extensions of global function fields. Furthermore we afford the code of the Magma scripts to verify the results and construct more examples in similar fashion.
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