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

Abstract

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\u200e. \u200eLet $mathcal{O}_q$ be the set of all odd order elements of $mathbb{F}_q^*$\u200e. \u200eThen $mathcal{O}_q$ turns up as a subgroup of $mathcal{S}_q$\u200e. \u200eIn this paper\u200e, \u200ewe show that $mathcal{O}_q=langle4rangle$ if $q=2t+1$ and\u200e, \u200e$mathcal{O}_q=langle trangle $ if $q=4t+1$\u200e, \u200ewhere $q$ and $t$ are odd primes\u200e. \u200eFurther\u200e, \u200ewe determine the coefficients of irreducible factors of $x^{2^nt}-1$ using generators of these special subgroups of $mathbb{F}_q^*$