Primitive idempotents of cyclic codes of length p and 2p

Let F-q be a finite field of order q with odd characteristic l and p be an odd prime such that gcd(p, q) = 1. Let the multiplicative order of q modulo p be phi(p)/4. Then the explicit expressions for primitive idempotents in the semi-simple ring F-q[x]/< x(P)-1 > are computed when p =1+4(ll')(2), p = 9+4(ll')(2), p = 9+4(ll' +/- 2)(2), where l' is an integer, using a simple approach. Further it is shown that in order to drive primitive idempotents in F-q[x]/< x(2P)-1 >, it is sufficient to obtain the exponential sums modulo p. Hence in cases p = 1+4(ll')(2), p = 9+4(ll')(2), p = 9 +4(ll' +/- 2)(2), explicit expressions for primitive idempotents of F-q[x]/< x(2P)- 1 > are also computed.