Factorization of composed polynomials and applications

TL;DR

This paper studies how to factor polynomials of the form f(x^n) over finite fields, generalizing previous results on binomials, and provides formulas for counting irreducible factors under certain conditions.

Contribution

It generalizes recent work on binomial factorization to a broader class of composed polynomials and derives explicit formulas for their irreducible factors.

Findings

Provides a general factorization method for f(x^n) over finite fields.

Derives explicit formulas for counting irreducible factors.

Extends previous results on binomials to more general polynomials.

Abstract

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a prime power and $n$ be a positive integer. In this paper, we explore the factorization of $f(x^{n})$ over $\mathbb{F}_q$, where $f(x)$ is an irreducible polynomial over $\mathbb F_q$. Our main results provide generalizations of recent works on the factorization of binomials $x^n-1$. As an application, we provide an explicit formula for the number of irreducible factors of $f(x^n)$ under some generic conditions on $f$ and $n$.