In Lang's Algebraic Number Theory book he uses a certain bound on the discriminant of the Galois closure of a number field $K$ without proof stating that it is an easy exercise. Let $\tilde{K}$ be the Galois closure of $K$. Set $[K:\mathbb{Q}] = N$ and $[\tilde{K}:\mathbb{Q}] = \tilde{N}$. Let us also denote the absolute values of the discriminants as $d_K$ and $d_{\tilde{K}}$. Then $d_{\tilde{K}} \leq d_{K}^{\frac{\tilde{N}}{2}}$.
What I tried was the following. Let $L$ be the subfields of $\tilde{K}$ such that $K$ and $L$ are linearly disjoint, and $\tilde{K} = KL$, then $[L:\mathbb{Q}] = \frac{\tilde{N}}{N}$. Now we have that $d_{\tilde{K}} = d_{K}^{\frac{\tilde{N}}{N}}d_{L}^{N}$.
But I am struggling to proceed further. I think we need to bound $d_L$. Any ideas would be extremely helpful. Thank you.