Explicit factorization of $x^{2^nd}-1$ over a finite field

TL;DR

This paper provides an explicit factorization of the polynomial $x^{2^nd}-1$ over finite fields of odd characteristic, specifically when $d$ divides $q+1$, enhancing understanding of polynomial structures in finite fields.

Contribution

It presents the first explicit factorization of $x^{2^nd}-1$ over finite fields for the case when $d$ is an odd divisor of $q+1$, filling a gap in polynomial factorization theory.

Findings

Explicit factorization formulas derived for $x^{2^nd}-1$

Applicable to finite fields of odd characteristic with specific divisibility conditions

Advances the understanding of polynomial structures over finite fields

Abstract

Let $\mathbb{F}_q$ be a finite field of odd characteristic containing $q$ elements and integer $n\ge 1$. In this paper, the explicit factorization of $x^{2^nd}-1$ over $\mathbb{F}_q$ is obtained when $d$ is an odd divisor of $q+1$.