L is a Galois extension and we want Galois group G to be generated by the inertial groups for all primes
Any suggestions?
Here is my proof (in process): The intermediate field $L_{I}$ for the inertia group over prime $p\in \mathbb{Q}$ is unramified $\Rightarrow p\nmid d_{\mathbb{L_{I}}}\Rightarrow [L_{I}:\mathbb{Q}]=1$. Also, every prime ideal in L lies over a rational prime