Field extension $\Bbb Q (\zeta_p)$

Question

This is for a quick verification.

Define the $p$th cyclotomic polynomial

$$\Phi_p(X) = \sum_{k = 0}^{p-1}X^k$$

where $p$ is an odd prime.

If I understand correctly, the minimal polynomial of $\zeta_p = e^{\frac{2\pi i}{p}}$ is $\Phi_p$ and so can we write

$$\Bbb Q (\zeta_p) \cong \Bbb Q[X]/(\Phi_p)$$

where $(\Phi_p)$ is the ideal generated by $\Phi_p$. If I am wrong, why am I wrong?

Answer 1

You are correct. It's worth stressing that the form you give is only correct for primes. The general case is discussed here.