  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.
