Non-existence of a particular type of tower of number fields

Question

I have number fields $\mathbb{Q}\subset K\subset H$ where $K\subset H$ is Galois. I want to show that is is impossible for a rational prime $p\in\mathbb{Z}$ to remain first inert in $K$ but then for $p\mathcal{O}_K$ to split in $H$.

My lecturer advised me to use:

1. The theorem that in an extension of number fields $K\subset L=K(\alpha)$, $f=\text{min}(\alpha,K)\in K[X]$: the set of primes over a fixed prime $P$ in $K$ bijects with the set of irreducible factors of $f$ as an element of $K_P[X]$. (Possibly a more detailed version of this.)

together with

2. Hensel's Lemma.

However I haven't been able to make progress with this at all. Any help greatly appreciated. I think this is very related: Ramification in a tower of extensions, since the result the accepted answer quotes is used in the proof of 1.