Some subgroups of a finite field and their applications for obtaining explicit factors

Let ${\mathcal{S}}_q$ denote the group of all square elements in the multiplicative group ${\mathbb{F}}_q^{\ast }$ 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 ${\mathbb{F}}_q^{\ast }$. Then ${\mathcal{O}}_q$ turns up as a subgroup of ${\mathcal{S}}_q$. In this paper, we show that ${\mathcal{O}}_q=⟨4⟩$ if $q=2t+1$ and, $$\mathcal{O}$_q=\langle t\rangle $if$q=4t+1$,where$q$and$t$ are odd primes. This paper also gives a direct method for obtaining the coefficients of irreducible factors of $x^{2^{n}t}-1$ in ${\mathbb{F}}_q[x]$ using the information of generator elements of ${\mathcal{S}}_q$ and ${\mathcal{O}}_q$, when $q$ and $t$ are odd primes such that $q=2t+1$ or $q=4t+1$.