When is a cyclotomic polynomial over a finite field a minimal polynomial?

Question

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

Answer 1

In general, a polynomial $f\in F[X]$ is the minimal polynomial of its roots over $F$ if and only if $f$ is irreducible over $F$. Cyclotomic polynomials provide no exception.

However, while cyclotomic polymomials are irreducible over $\mathbb Q$, not all of them are irreducible when considered as a polynomial over a finite field. For example, over $\mathbb F_2$ the $7$th cyclotomic polynomial $\Phi_7 = X^6 + X^5 + X^4 + X^3 + X^2 + X + 1$ factors into two factors of degree $3$: $$ \Phi_7 = (X^3 + X + 1)(X^3 + X^2 + 1) $$ Hence $\Phi_7\in\mathbb F_2[X]$ is not irreducible and therefore not the minimal polynomial of its roots.

Note: This was written as a response to an earlier version of the question.

Answer 2

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 $\mathbf Q$!