class group size of cyclotomic field subextension

Question

In the following, let $\mathbb{Q_1}$ denote the subfield of degree $p$ over $\mathbb{Q}$ in the $p^2$- cyclotomic extension.

What is the best known upper bound for the size of its class group, $\text{Cl}(\mathbb{Q_1})$?

It is known that the bounds of the order of $p^{(p-1)/2}$ are known for the class number of the $p$-cyclotomic fields. I was wondering if something similar is known for $Q_1$. For instance, if some bound in the order of $O(p^{p/2})$ or at least $O(p^p)$ can be proved.

Answer 1

The $p$-primary part of the class group of $\mathbb{Q}_1$, and in fact all $\mathbb{Q}_n$ in the cyclotomic tower of $\mathbb{Q}$, is trivial for all $p$. This is contained in Proposition 13.22 of Washington's "Introduction to cyclotomic fields".