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, and qf=1+pλ. Here expressions for all the primitive idempotents in the ring Rpn=GF(q)[x]/(xpn−1), for any positive integer n, are obtained in terms of cyclotomic numbers, provided p does not divide λ if n 2. The dimension, generating polynomials and minimum distances of minimal cyclic codes of length pn over GF(q) are also discussed.