For the first glance it should follow form the Abhaynkar's lemma (see "Algebraic Function Fields" by Stichtenoth, Theorem 3.9.1) and the fact the Galois closure is the composite of all the different embeddings of L over K into fixed algebraic closure of K (so each of them has the same properties of tame ramifications). Then we just apply the lemma and get the result that p=char(K) $p=char(K)$ does not divide e_P $e_P$ for any place P in K.