Checking irreducibility of a degree $6$ polynomial over $\mathbb{F}_3$

Question

I want to check whether the following polynomial is irreducible over $\mathbb{F}_3$: $$f(X)=X^6+X^5+X^4+X^3+X^2+X+1$$

One obvious way to solve the problem is to divide $f(X)$ by irreducible polynomials of smaller degree.

My question:

Is there any other clever way to check whether this polynomial is irreducible or not?

Answer 1

It's a cyclotomic polynomial. Its zeros are the seventh roots of unity over $\Bbb F_3$. But these generate the field $\Bbb F_{3^k}$ where $3^k$ is the least power of $3$ with $3^k\equiv1$ (mod~$7$). The polynomial is irreducible iff $k=6$. Is it?