M\"obius Polynomials

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.