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

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

1. 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.

• I'm not sure this is true as stated. For example, $7$ is inert in $K=\mathbb Q(i)$, but $7\mathbb Z[i]$ splits in $K(\sqrt {-6})$ since $7=(1+\sqrt{-6})(1-\sqrt{-6})$. Have I misunderstood something? – Mathmo123 May 24 '16 at 8:29
• You will need the assumption that K/{\mathbb Q}\$ be cyclic, not just Galois. – franz lemmermeyer May 26 '16 at 7:41