SOME SUBGROUPS OF F * q an‎d EXPLICIT FACTORS OF x2nd - 1 2 Fq[x]

Let Sq denote the group of all square elements in the multiplicative group F * q of a nite eld 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 x2nt - 1 using generators of these special subgroups of F * q .