Cyclotomic numbers and primitive idempotents in the ring GF(q)[x]/(xpn-1)

Let q be an odd prime power and p be an odd prime with gcd(p,q)=1. Let order of q modulo p be f, gcd(p-1f,q)=1 and q^f=1+p@l. Here expressions for all the primitive idempotents in the ring R"p"^"n=GF(q)[x]/(x^p^^^n-1), for any positive integer n, are obtained in terms of cyclotomic numbers, provided p does not divide @l if n>=2. The dimension, generating polynomials and minimum distances of minimal cyclic codes of length p^n over GF(q) are also discussed.