Genus fields of congruence function fields

Let k be a rational congruence function field and consider a finite separable extension K/k. We consider the extension K/k satisfying the following condition. For each prime in k at least one prime in K above it is tamely ramified. Then, except for constants, we find the genus field Kge of K/k. In general, we describe the genus field of a global function field. We present some applications and examples of the main results.