Define the mth cyclotomic polynomial $\displaystyle \Phi_m = \prod_{i \in U(m)} (X - \xi^i)$, where $ \xi$ is a primitive mth root of unity.
Why is true that $X^m - 1 = \displaystyle \prod_{d|m} \Phi_d$?
I can see that $X^m -1 = \displaystyle \prod_{i \in \mathbb Z / m \mathbb Z} (X - \xi^i)$ where $\xi$ is a primitive mth root of unity, but I can't see why this is equal to $ \displaystyle \prod_{d|m} \Phi_d$. The degrees of these two polynomials are the same, but that's about all I'm managing to notice...
Any help appreciated. Thanks