Composed products and factors of cyclotomic polynomials over finite fields

Let q = p (s) be a power of a prime number p and let be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over . In particular we obtain the explicit factorization of the cyclotomic polynomial over where both r a parts per thousand yen 3 and q are odd, gcd(q, r) = 1, and . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of over is given to us as a known. Let be the factorization of into powers of distinct primes p (i) , 1 a parts per thousand currency sign i a parts per thousand currency sign s. In the case that the multiplicative orders of q modulo all these prime powers are pairwise coprime, we show how to obtain the explicit factors of from the factors of each . We also demonstrate how to obtain the factorization of from the factorization of when q is a primitive root modulo m and Here is the Euler's totient function, and ord (n) (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over and generalize a result due to Varshamov (Soviet Math Dokl 29:334-336, 1984).