Fekete polynomials, quadratic residues, and arithmetic

TL;DR

This paper explores Fekete polynomials linked to prime numbers, analyzing their special values, Galois groups, and proposing a conjecture on their algebraic structure, connecting quadratic residues with class numbers and Bernoulli numbers.

Contribution

It introduces two new related polynomials, expresses their special values in terms of class numbers and Bernoulli numbers, and studies their Galois groups with a conjecture on their structure.

Findings

Expressed special values in terms of class numbers and Bernoulli numbers

Analyzed splitting fields and Galois groups of the polynomials

Proposed a conjecture on the Galois group structure

Abstract

Fekete polynomials associate with each prime number $p$ a polynomial with coefficients $-1$ or $1$ except the constant term, which is 0. These coefficients reflect the distribution of quadratic residues modulo $p$. These polynomials were first considered in the 19th century in relation to the studies of Dirichlet $L$-functions. In our paper, we introduce two closely related polynomials. We then express their special values at several integers in terms of certain class numbers and generalized Bernoulli numbers. Additionally, we study the splitting fields and the Galois group of these polynomials. In particular, we propose a conjecture on the structure of these Galois groups.