# Number of irreducible polynomials whose compositions with monic monomials have large irreducible factors

## TL;DR

This paper counts monic irreducible polynomials over finite fields that, when composed with monomials, contain large irreducible factors, aiding the analysis of randomized algorithms in matrix group computations.

## Contribution

It provides a formula for the number of such polynomials with specific degree and irreducibility properties, advancing understanding in finite field polynomial structures.

## Key findings

- Derived explicit counts for polynomials with specified irreducibility properties.
- Established conditions under which these polynomials contain large irreducible factors.
- Applied results to justify randomized algorithms in matrix group theory.

## Abstract

Given a prime power $q$ and positive integers $m,t,e$ with $e > mt/2$, we determine the number of all monic irreducible polynomials $f(x)$ of degree $m$ with coefficients in $\mathbb{F}_q$ such that $f(x^t)$ contains an irreducible factor of degree $e$. Polynomials with these properties are important for justifying randomised algorithms for computing with matrix groups.