Enumeration of Cyclic Codes over GF(19)

In this paper we seek the number of irreducible polynomials of x^n-1 over GF (19). First, we factorize x^n-1 into irreducible polynomials over GF (19) using cyclotomic cosets of 19 modulo n. The number of irreducible polynomial factors of x^n-1 over GF (19) is equal to the number of cyclotomic cosets of 19 modulo n and each monic divisor of x^n-1 is a generator polynomial of a cyclic code in GF (19). Next, we show that the number of cyclic codes of length n over a finite field GF (19) is equal to the number of polynomials that divide x^n-1. Lastly, we enumerate the number of cyclic codes of length n, for 1n20 and when n=19k, n=19^k for 1k20