The genus fields of Artin–Schreier extensions

Let q be a power of a prime number p. Let k=Fq(t) be the rational function field with constant field Fq. Let K=k(α) be an Artin–Schreier extension of k. In this paper, we explicitly describe the ambiguous ideal classes and the genus field of K. Using these results, we study the p-part of the ideal class group of the integral closure of Fq[t] in K. We also give an analogue of the Rédei–Reichardt formula for K.