Prime Splits Completely in Every Intermediate Field

Question

Suppose I have a finite field extension of number fields (finite field extensions over $\mathbb{Q}$), say $K\subset L$. Say $P$ is a prime in the number ring contained in $K$ such that $P$ splits completely in every intermediate field strictly between $K$ and $L$, but does not split completely in $L$. I have already shown that the Galois group of such a field has order that is a prime power of a prime $p$ and that it contains a unique smallest normal subgroup of order $p$.

Now, I just want to find an example of such a field extension $L$ over $K$ where $K = \mathbb{Q}$ and the Galois group of $L$ over $K$ is of order $p^{k}$ for $k>1$. Any suggestions?

Answer 1

Let $\zeta_5 = e^{2\pi i/5}$ be a $5$-th root of unity. Then $L = \Bbb Q(\zeta_5)$ is an abelian extension with Galois group $$(\Bbb Z/5 \Bbb Z)^\times = \Bbb Z/4 \Bbb Z.$$ So we have one intermediate extension corresponding to the one non-trivial subgroup of $\Bbb Z / 4 \Bbb Z$. It is $\Bbb Q(\sqrt{5})=M$, because $\zeta_5+\zeta_5^{-1} = (\sqrt{5}-1)/2 \in \Bbb Q(\zeta_5).$

An odd prime $p$ is completely split in $\Bbb Q(\sqrt{5})=M$ iff $\left(\frac{p}{5} \right)=1$, because $d_M = 5$.

An odd prime $p$ is completely split in $\Bbb Q(\zeta_5)$ iff $p \equiv 1 \bmod 5$.

So choose any prime $p \equiv 4 \bmod 5$ to get an example, e.g. $19 = (8+3\sqrt{5})(8-3\sqrt{5})$.