The study of cyclic codes via cyclotomic polynomial over Galois field has been active area of research due to their direct application in generating cyclic codes and error-correcting codes. Let q be a prime number and Fq be a given finite field with q elements. This research investigates the cyclotomic polynomial yn −1 specifically focusing on cases where yn −1 completely decomposes into linear factors over Fq for q ≤ 37 and n ≥ 2. The relationships between the field, the sum, and the product of the zeros of these linear factors are explored. The results shows that for each tested pair (q,n) where n | (q−1), the sum of the roots is always ≡ 0 (mod q), the product of the roots is ≡ −1 (mod q) and the inverse of the ratio of the product of the roots to n is ≡ q−n (mod q). The predictable modular relationships among the zeros, can be applied to the efficient design of generator polynomials with desired properties.
Ondiany et al. (Fri,) studied this question.