  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.
