Sequences of irreducible polynomials without prescribed coefficients over odd prime fields

TL;DR

This paper constructs infinite sequences of monic irreducible polynomials over odd prime fields using Cohen's transformation, without restrictions on initial coefficients, and describes their degree growth pattern.

Contribution

It introduces a method to generate infinite irreducible polynomial sequences over odd prime fields without coefficient restrictions, expanding previous approaches.

Findings

Sequences are infinite under specified conditions.

Degree doubling occurs after initial segment.

No assumptions on initial polynomial coefficients.

Abstract

In this paper we construct infinite sequences of monic irreducible polynomials with coefficients in odd prime fields by means of a transformation introduced by Cohen in 1992. We make no assumptions on the coefficients of the first polynomial $f_0$ of the sequence, which belongs to $\F_p [x]$, for some odd prime $p$, and has positive degree $n$. If $p^{2n}-1 = 2^{e_1} \cdot m$ for some odd integer $m$ and non-negative integer $e_1$, then, after an initial segment $f_0, ..., f_s$ with $s \leq e_1$, the degree of the polynomial $f_{i+1}$ is twice the degree of $f_i$ for any $i \geq s$.