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

Document Type : Research Paper


Department of‎ ‎Mathematics, ‎Deenbandhu Chhotu Ram University of Science and Technology, Murthal-131039‎, ‎Sonepat‎, ‎India


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 $\mathcal{O}_q=\langle4\rangle$ if $q=2t+1$ and‎, ‎$\mathcal{O}_q=\langle t\rangle $ 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 $\mathbb{F}_q^*$