Decomposition of places in dihedral and cyclic quintic trinomial extensions of global fields

In this paper, we give a complete and explicit description of the splitting behavior of any place in a quintic trinomial dihedral or cyclic extension of a rational function field of finite characteristic distinct from 2 and 5. Our characterization depends only on the order of the base field and a parametrization of the coefficients of the generating trinomial. Moreover, we contrast some of our results to trinomial dihedral number fields of prime degree, where the unit rank behaves quite differently from the function field scenario.