Can somebody help me with this algebra problem?

[Vancouver]

Let n and k be relatively prime positive integers. Let f(X) and g(X) be polynomials with integers coefficients such that
f(X)g(X) = X^n - 1.
Prove that
f(X^k) / f(X)
is a polynomial with integer coefficients.

Example:
n=6 and k=5
f(X) = X^2 - X + 1
g(X) = X^4 + X^3 - X - 1
We now have
f(X)g(X) = X^6 - 1
and you can verify that f(X^5)/f(X) has integer coefficients.

[Vancouver]

I was being a bit facetious. The claim follows easily enough from a theorem in advanced algebra (the theorem that all cyclotomic polynomials are irreducible), but an independent proof of this claim would imply the other theorem, and that would be nice because the only known proofs of the latter are a little bit unnaturally long and contrived.

[pilobolus]

I'd love to help, but I am a Bush League academic...