Carlitz-Hayes plus Anderson's epsilon

Let k be the field of rational functions over the finite field of q elements. Let k(ac) be an algebraic closure of k. The maximal abelian extension k(ab) of k in k(ac) and its Galois group over k have been described in the obvious manner by Hayes by developing ideas of Carlitz. Let k(ab+epsilon) be the compositum of all sub fields of k(ac) which are Kummer (q-1)-extensions over k(ab) and are Galois over k. In this paper we exhibit an explicit description of the field k(ab+epsilon) and the Galois group of k(ab+epsilon/k). Our main motivation is the recent work of Anderson about the similar question over the rational number field.