On period polynomials of degree $2^m$ for finite fields

TL;DR

This paper derives explicit factorizations of reduced period polynomials of degree 2^m over finite fields with specific characteristics, extending previous results for smaller degrees.

Contribution

It generalizes the explicit factorization of period polynomials to higher degrees 2^m for certain finite fields, building on prior work for lower degrees.

Findings

Explicit factorizations for degree 2^m polynomials

Extension of previous results to larger degrees

Applicable to finite fields with characteristic p ≡ 3 or 5 mod 8

Abstract

We obtain explicit factorizations of reduced period polynomials of degree $2^m$, $m\ge 4$, for finite fields of characteristic $p\equiv 3$ or $5\pmod{8}$. This extends the results of G. Myerson, who considered the cases $m=1$ and $m=2$, and S. Gurak, who studied the case $m=3$.