Abstract

We present new results about the factorization of $\Phi_p(M) \in \mathbb{F}_2[x]$, where $p$ is a prime number, $\Phi_p$ is the corresponding cyclotomic polynomial, and $M$ is a Mersenne prime polynomial. In particular, these results improve our understanding of the factorization of the sum of the divisors of $M^{2h}$ for a positive integer $h$. This is related to the fixed points of the sum of divisors function $\sigma$ on $\mathbb{F}_2[x]$. The factorization of composed polynomials over finite fields is not well understood, and classical results on cyclotomic polynomials primarily concern the special case where $M$ is replaced by $x$.

Keywords: Cyclotomic polynomials; sum of divisors; finite fields.

MSC: 11T55, 11T06

DOI: 10.57016/MV-HMLV9f07

Pages:  1--8