I am just beginning my study for field theory and I am trying to solve a problem:
Let $p$ be a prime and $\zeta$ be the primitive $p$th root of unity. Then calculate $[\mathbb{Q}(\sqrt[p]{2},\zeta) : \mathbb{Q}]$.
Here is my attempt:
Since we have $\mathbb{Q} \subseteq \mathbb{Q}(\sqrt[p]{2}) \subseteq \mathbb{Q}(\sqrt[p]{2},\zeta)$, we get
$$[\mathbb{Q}(\sqrt[p]{2},\zeta) : \mathbb{Q}] = [\mathbb{Q}(\sqrt[p]{2},\zeta) : \mathbb{Q}(\sqrt[p]{2})] [\mathbb{Q}(\sqrt[p]{2}) : \mathbb{Q}]$$
And this means we only need to calculate $[\mathbb{Q}(\sqrt[p]{2},\zeta) : \mathbb{Q}(\sqrt[p]{2})]$ and $[\mathbb{Q}(\sqrt[p]{2}) : \mathbb{Q}]$.
For $[\mathbb{Q}(\sqrt[p]{2}) : \mathbb{Q}]$, the minimal polynomial of $\sqrt[p]{2}$ over $\mathbb{Q}$ is $f(x) = x^p - 2$, and this means
$$[\mathbb{Q}(\sqrt[p]{2}) : \mathbb{Q}] = \deg(x^p - 2) = p$$
For $[\mathbb{Q}(\sqrt[p]{2},\zeta) : \mathbb{Q}(\sqrt[p]{2})]$, a basis of $\mathbb{Q}(\sqrt[p]{2},\zeta)$ over $\mathbb{Q}(\sqrt[p]{2})$ is $\{1, \zeta, \zeta^2,...,\zeta^{p-1}\}$, so we have
$$[\mathbb{Q}(\sqrt[p]{2},\zeta) : \mathbb{Q}(\sqrt[p]{2})] = \mid\{1, \zeta, \zeta^2,...,\zeta^{p-1}\} \mid = p$$
Therefore we get
$$[\mathbb{Q}(\sqrt[p]{2},\zeta) : \mathbb{Q}] = [\mathbb{Q}(\sqrt[p]{2},\zeta) : \mathbb{Q}(\sqrt[p]{2})] [\mathbb{Q}(\sqrt[p]{2}) : \mathbb{Q}] = p^2$$
$\textbf{Question:}$ Can someone help me check whether my attempt is right or not? I am just beginning my study and I am really not confident in this. If someone is willing to help, your help will be greatly appreciated! Thanks!
