class number divisibility in $p$-extensions

In this entry, the class number of a number field $F$ is denoted by $h_F$.

Theorem 1.

Let $p$ be a fixed prime number.

• Let $F/K$ be a Galois extension with Galois group $\mathrm{Gal}(F/K)$ and suppose $F/K$ is a $p$-extension (so $\mathrm{Gal}(F/K)$ is a $p$-group). Assume that there is at most one prime or archimedean place which ramifies in $F/K$. If $h_F$ is divisible by $p$ then $h_K$ is also divisible by $p$.
  

• Let $F/\mathbb{Q}$ be a Galois extension of the rational numbers and assume that $\mathrm{Gal}(F/\mathbb{Q})$ is a $p$-group and at most one place (finite or infinite) ramifies then $h_F$ is not divisible by $p$.