PreprintarXiv licencearXiv
Cyclotomic polynomials with trinomial irreducible factors over finite fields
Abstract
We classify the pairs (q,n), where q is a prime power, n>1, and gcd(n,q)=1, for which every irreducible factor of Φn(x) over Fq has exactly three nonzero terms. The positive support exponents of these factors have a common greatest common divisor, which gives a unique reduction by a power substitution. The reduced degree is two in odd characteristic and belongs to {2,3,4} in characteristic two; the cubic and quartic cases arise by scalar substitution from the layers of orders $7$ and $15$. This yields necessary and sufficient arithmetic conditions on (q,n) and explicit factorizations. Among the classified families, precisely those arising from quartic bases have a divisor layer containing an irreducible factor with more than three terms.
Related
- Post-critically finite polynomials over cyclotomic closures of number fields
- Monogenity of the irreducible factors of (k,t)-Fibonacci and (k,t)-Lucas polynomials
- Invariant factors from products of cyclotomic polynomials, with applications
- Scott-type inequalities for trinomial h*-polynomials
- Cyclotomic Euler–Mahonian polynomials
BibTeX
@misc{zhu2026cyclotomic,
title = {{Cyclotomic polynomials with trinomial irreducible factors over finite fields}},
author = {Li Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00647},
eprint = {2611.00647},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00647},
}
RIS
TY - UNPB
TI - Cyclotomic polynomials with trinomial irreducible factors over finite fields
AU - Li Zhu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00647
UR - https://arxiv.org/abs/2611.00647
AB - We classify the pairs (q,n), where q is a prime power, n>1, and gcd(n,q)=1, for which every irreducible factor of Φ_n(x) over F_q has exactly three nonzero terms. The positive support exponents of these factors have a common greatest common divisor, which gives a unique reduction by a power substitution. The reduced degree is two in odd characteristic and belongs to {2,3,4} in characteristic two; the cubic and quartic cases arise by scalar substitution from the layers of orders $7$ and $15$. This yields necessary and sufficient arithmetic conditions on (q,n) and explicit factorizations. Among the classified families, precisely those arising from quartic bases have a divisor layer containing an irreducible factor with more than three terms.
ER -