Let p be any odd prime number, m and s be arbitrary positive integers, and let F p m be the finite field of cardinality p m. Existing literature only determines the number of all (Euclidean) self-dual cyclic codes of length p s over finite chain ring R= F p m+ u F p m (u 2= 0), such as Dinh et al.(2018). Using some combinatorial identities, we obtain certain properties for Kronecker product of matrices over F p with a specific type. On that basis, we give an explicit representation and enumeration for all distinct self-dual cyclic codes of length p s over R. Moreover, we provide an efficient method to construct every self-dual cyclic code of length p s over R precisely.