Some subgroups of $\mathbb{F}_q^*$ and explicit factors of $x^{۲^nd}-۱\in\mathbb{F}_q[x]$

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 $\mathcal{O}_q=\langle۴\rangle$ if $q=۲t+۱$ and‎, ‎$\mathcal{O}_q=\langle t\rangle $ if $q=۴t+۱$‎, ‎where $q$ and $t$ are odd primes‎. ‎Further‎, ‎we determine the coefficients of irreducible factors of $x^{۲^nt}-۱$ using generators of these special subgroups of $\mathbb{F}_q^*$

Keywords:

‎Polynomials over finite fields‎ , ‎Cyclotomic polynomials‎ , ‎Special groups