TITLE: When is a cyclotomic polynomial over a finite field a minimal polynomial?
QUESTION [12 upvotes]: When is the cyclotomic polynomial $f(x)$ over a finite field $\mathrm{F}_q$ also the minimal polynomial of some element $\alpha \in \mathrm{F}_q$?

REPLY [17 votes]: One can show that the cyclotomic polynomial $\Phi_n(X)$ is irreducible over $\mathbf F_p$ precisely when $p$ has multiplicative order $\varphi(n)$ modulo $n$. This follows from the theory of cyclotomic extensions of $\mathbf Q$.
In particular, if $(\mathbf Z/n\mathbf Z)^\times$ is not cyclic (i.e. unless $n$ is an odd prime power, twice an odd prime power, or $n=2$ or $4$), then $\Phi_n(X)$ is reducible modulo every prime $p$, despite being irreducible over $\mathbf Q$!