Fekete polynomials of principal Dirichlet characters

TL;DR

This paper investigates Fekete polynomials linked to principal Dirichlet characters, focusing on their cyclotomic factors, irreducibility, and Galois groups, combining theoretical analysis with computational evidence.

Contribution

It introduces a new variant of Fekete polynomials, analyzes their cyclotomic factors, and explores their irreducibility and Galois groups, especially when the modulus is a product of two primes.

Findings

Cyclotomic factors of the polynomials are characterized and conjecturally fully described.

The non-cyclotomic part appears always irreducible in computational experiments.

The Galois groups of these polynomials are identified for small cases, raising new questions.

Abstract

Fekete polynomials associated to quadratic Dirichlet characters have interesting arithmetic properties, and have been studied in many works. In this paper, we study a seemingly simpler yet rich variant: the Fekete polynomial $F_n(x) = \sum_{a=1}^n \chi_n(a) x^a$ associated to a principal Dirichlet character $\chi_n$ of modulus $n$. We investigate the cyclotomic factors of $F_n$ and conjecturally describe all of them. One interesting observation from our computations is that the non-cyclotomic part $f_n$ of $F_n(x)/x$ seems to be always irreducible. We study this factor closely in the special case that $n$ is a product of two odd primes, proving separability in specific cases, and studying its coefficients and special values. Combining these theoretical results with computational evidence lets us identify the Galois group of $f_n$ for small $n$, and raises precise questions in general.