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

2019/03/26 by Pannek, Sabina B.
#11T06 #12E05 #20P05 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1903.10846

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 \mathbbF_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.

Related

• Irreducibility results for compositions of polynomials in several variables
• On the distribution of polynomials having a given number of irreducible factors over finite fields
• Irreducibility of Random Polynomials
• Factorization of a class of composed polynomials
• Factoring polynomials of the form f(xⁿ)∈ \mathbbF_q[x]