How to prove $[\mathbb Q(\zeta_{p^k}): \mathbb Q] = (p-1)*p^{k-1}$?

Question

I want to prove that $[\mathbb Q(\zeta_{p^k}): \mathbb Q] = (p-1)*p^{k-1}$ where $\zeta_p = \exp\left (\frac {2 \pi i} p\right )$. I have done this for $k = 2$ by looking at $m(x) = \frac{X^p-1}{X-1}$ and $n(x) = m(x^p)$ which are irreducible polynomials over $\mathbb Q$. This gives the degree for $ k=1$ and $k=2$. Can I somehow prove my assertion by induction of must I proceed like $n'(x) := m(x^{p^2})$ etc. ?