i was wondering if anyone could give me a quick explanation about a problem concerning the degree of a field extension $Z\mid \mathbb{Q}$. The problem is as follows :
Let $Z\subset \mathbb{C}$ be the splitting field of $f(X)=X^7-3$. Also let $\alpha = 3^{\frac{1}{7}}$ and $\zeta=e^{\frac{2\pi i}{7}}$. Calculate $[Z : \mathbb{Q}]$.
I know the Polynomial $f$ to be irreducible because of Eisenstein (and also about the product rule for the degree of extension fields), but every time the field extension is slightly more complicated than simply adjoining the square root of a prime to $\mathbb{Q}$, i get confused because i cant simply say $[Z : \mathbb{Q}]=7$, because that would leave out the roots that have non zero imaginary parts(?). What exactly should my thought process be in these types of exercises?
Hint:
The roots of your polynomial are $\;\alpha,\,\alpha\zeta,\ldots,\alpha\zeta^6\;$, yet observe that in fact $\;Z=\Bbb Q(\alpha,\,\zeta)\;$, and thus for its degree you can try
$$|Z:\Bbb Q|=\left|\Bbb Q(\alpha,\zeta):\Bbb Q(\zeta)\right|\,\left|\Bbb Q(\zeta):\Bbb Q\right|$$
Can you get it from here? Try to check stuff about irreducible polynomials and etc.