Explicit factorization of $x^n-1\in \mathbb F_q[x]$

TL;DR

This paper provides explicit factorizations of the polynomial x^n - 1 over finite fields into irreducible binomials and certain trinomials, under specific conditions on q and n, enhancing understanding of polynomial structure in finite fields.

Contribution

It introduces explicit factorizations of x^n - 1 into irreducible binomials and trinomials under certain conditions, extending previous results on polynomial factorization over finite fields.

Findings

Explicit factorization into irreducible binomials x^t - a.

Extension to factors of the form x^{2t} - a x^t + b.

Conditions on q and n for the factorizations to hold.

Abstract

Let $\mathbb F_q$ be a finite field and $n$ a positive integer. In this article, we prove that, under some conditions on $q$ and $n$, the polynomial $x^n-1$ can be split into irreducible binomials $x^t-a$ and an explicit factorization into irreducible factors is given. Finally, weakening one of our hypothesis, we also obtain factors of the form $x^{2t}-ax^t+b$ and explicit splitting of $x^n-1$ into irreducible factors is given.