Let $K$ be an finite abelian extension of $\mathbf{Q}$ conductor $p$, where $p$ is an odd prime. That is, $K \subset \mathbf{Q}(\mu_ p)$, the $p$-th cyclotomic field. Let $h_K$ be the class number of $K$.

If $[K:\mathbf{Q}]=2$, we know that $K=\mathbf{Q}(\sqrt{p})$ or $\mathbf{Q}(\sqrt{-p})$. Gauss's genus theory tells us that $h_K\equiv 1 \pmod 2$.

If $[K:\mathbf{Q}]=3$, a paper of G.Gras https://eudml.org/doc/151629 (Page 94 line 1-2) says that $h_K \equiv 1 \pmod 3$. Unfortunately, I did not find the proof of this result. I am very appreciated if someone gives references or ideas.

So my question is that

if $[K:\mathbf{Q}]=n$ is a prime number, then do we have that $h_K \equiv 1 \pmod n$?