We discuss cyclic and negacyclic codes of length $p^s$ over Galois ring $GR(p^a,m)$. The ambient rings for those codes are chain rings if and only if the codes are negacyclic and $p=2$. In this case, complete structure and Hamming distances of all such negacyclic codes are established. For all other cases, it is shown that the ambient rings are local ring with the non-principal maximal ideal generated by $2$ elements. We establish the nilpotency indexes of such 2 elements in the more general setting where the alphabet is any finite chain ring.