  We introduce the M\"obius polynomial $ M_n(x) = \sum_{d|n} \mu\left( \frac nd
\right) x^d $, which gives the number of aperiodic bracelets of length $n$ with
$x$ possible types of gems, and therefore satisfies $M_n(x) \equiv 0$ (mod $n$)
for all $x \in \mathbb Z$. We derive some key properties, analyze graphs in the
complex plane, and then apply M\"obius polynomials combinatorially to juggling
patterns, irreducible polynomials over finite fields, and Euler's totient
theorem.
