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

TL;DR

This paper characterizes certain subgroups of finite field multiplicative groups and introduces a direct method for factoring specific polynomials over finite fields using subgroup generators.

Contribution

It explicitly describes the structure of subgroups of square and odd order elements in finite fields of odd characteristic and applies this to polynomial factorization.

Findings

_q=_q^2 if q=2t+1, with _q=_q^2 generated by 4.

_q=\u1d4b_t if q=4t+1, with _q=_q^2 generated by t.

Provides a method to obtain coefficients of irreducible factors of x^{2^nt}-1 using subgroup generators.

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. 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. This paper also gives a direct method for obtaining the coefficients of irreducible factors of $x^{2^nt}-1$ 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 $q=4t+1$.