Some subgroups of $mathbb{F}_q^*$ and explicit factors of $x^{2^nd}-1 in mathbb{F}_q[x]$

Let $mathcal{S}_q$ denote the group of all square elements in the multiplicative group $mathbb{F}_q^*$ of a finite field $mathbb{F}_q$ of odd characteristic containing $q$ elements‎. ‎Let $mathcal{O}_q$ be the set of all odd order elements of $mathbb{F}_q^*$‎. ‎Then $mathcal{O}_q$ turns up as a subgroup of $mathcal{S}_q$‎. ‎In this paper‎, ‎we show that $mathcal{O}_q=langle4rangle$ if $q=2t+1$ and‎, ‎$mathcal{O}_q=langle trangle $ 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 $mathbb{F}_q^*$