SOME SUBGROUPS OF F^∗q AND EXPLICIT FACTORS OF x^2nd − 1 ∈ Fq[x]

Let sq denote the group of all square elements in the multiplicative group f ∗ q of a finite field fq of odd characteristic containing q elements. let oq be the set of all odd order elements of f^∗ q . then oq turns up as a subgroup of sq. in this paper, we show that oq = ⟨4⟩ if q = 2t+ 1 and, oq = ⟨t⟩ if q = 4t+ 1, where q and t are odd primes. further, we determine the coefficients of irreducible factors of x^2^nt − 1 using generators of these special subgroups of f^∗ q .