abstract algebra - When does a polynomial divide $x^k - 1$ for some $k$?

Given a monic polynomial $f\in\mathbb{Z}[x]$, how can I determine whether there is a $k\in\mathbb{Z}^+$ such that $f|(x^k-1)$?

For example, $x^2-x+1$ divides $x^6-1$, but $x^2-x-1$ does not divide any such $x^k-1$ (unless I miss my mark!).

I would also be interested in finding how to answer this for other parameterized families of polynomials, or working over $\mathbb{Q}[x]$—I expect the former to be hard and the latter easy.

Answer

There are various algorithms known for such, based on properties of roots of unity, e.g.

See for example the following papers.

F. Beukers , C. J. Smyth. Cyclotomic Points on Curves, Proc. Milennial Conference on Number Theory, May 21–26, 2000, Urbana-Champaign, AK Peters (2001). (Section 2)

Iskander Aliev, Chris Smyth. Solving algebraic equations in roots of unity. (Section 2.1)

R. J. Bradford and J. H. Davenport, Effective tests for cyclotomic polynomials.
Symbolic and Algebraic Computation, Lecture Notes in Computer Science, 1989,
Volume 358/1989, 244-251, DOI: 10.1007/3-540-51084-2_22

Abstract (from Bradford and Davenport)

We present two efficient tests that determine if a given polynomial is cyclotomic, or is a product of cyclotomics. The first method uses the fact that all the roots of a cyclotomic polynomial are roots of unity, and the second the fact that the degree of a cyclotomic polynomial is a value of $\:\phi(n),$ for some $n$. We can also find the cyclotomic factors of any polynomial.

Here is the first method: