  We observe that the necklace polynomials $M_d(x) = \frac{1}{d}\sum_{e\mid
d}\mu(e)x^{d/e}$ are highly reducible over $\mathbb{Q}$ with many cyclotomic
factors. Furthermore, the sequence $\Phi_d(x) - 1$ of shifted cyclotomic
polynomials exhibits a qualitatively similar phenomenon, and it is often the
case that $M_d(x)$ and $\Phi_d(x) - 1$ have many common cyclotomic factors. We
explain these cyclotomic factors of $M_d(x)$ and $\Phi_d(x) - 1$ in terms of
what we call the \emph{$d$th necklace operator}. Finally, we show how these
cyclotomic factors correspond to certain hyperplane arrangements in finite
abelian groups.
