Ramification in Galois Closure

Question

I am trying to solve the following: Let $\Omega$ be the Galois Closure of $\mathbb{Q}[\sqrt[3]{2}]$ over $\mathbb{Q}$. Let $P$ be a prime ideal in $O_\Omega$. What are the possible values of $e(P|P\cap \mathbb{Z})$ and $f(P|P\cap \mathbb{Z})$ ($e$ and $f$ being the ramification index and inertial degree)? $$ $$ I manged to figure out the possible values for ideals in $O_{\mathbb{Q}[\sqrt[3]{2}]}$ using the discriminant, but what can I say about the Galois Closure?

Answer 1

Keith Conrad's notes The splitting field of $x^3-2$ over $\mathbb{Q}$ explain everything about this, see Theorem $5$. Your $\Omega$ is denoted by $K=\mathbb{Q}(\sqrt[3]{2},\omega)$ there.