Recently Anderson described explicitly the epsilon extension of the maximal abelian Q(ab) of the rational number field Q, which is the compositum of all subfield of C quadratic over Q(ab) and Galois over Q. We have given an analogous description of the epsilon extension of the maximal abelian extension of the rational function field over a finite field. In this paper we extend the theory of epsilon extensions partially to a global function field with a distinguished place. The new ingredient is that the base field may have non-trivial class group.