I have to show: For any number field $K$, there are infinitely many prime numbers $p \in \mathbb{N}$, that are totally split in $K$.
I think have already shown (with some hints my professor gave) that for any prime $p$ coprime to $ [K:\mathbb{Q}] $that $ \mathcal{O}_K/p\mathcal{O}_K \cong \mathbb{F}_p $ (for the integral closure $\mathcal{O}_K$ of $\mathbb{Z}$ in $K$).
So I have to show, that there are prime ideals $\mathfrak{p}_1, \dots ,\mathfrak{p}_{[K:\mathbb{Q}]} $ with $(p)=\mathfrak{p}_1...\mathfrak{p}_{[K:\mathbb{Q}]}$
But from here on I don't really know what to do.
Another hint was to use this fact: For any $f \in \mathbb{Z}[x] \setminus \mathbb{Z}$ there are infinitely many primes $p$ with $p \mid f(n)$ for some $n \in \mathbb{N}$.
also it should be useful to reduce the problem to a galois field
If anybody has an idea, I'd be glad to hear it
Thank you!