TITLE: $p$-torsion of class groups
QUESTION [7 upvotes]: Let $p$ be a fixed odd prime and $\ell$ be another prime such that $\ell \equiv 1 \pmod{p}$.
Consider the number field $\mathbb{Q}(\zeta_p)$ and its extension $\mathbb{Q}(\zeta_p, \zeta_\ell)$. Note that $\ell$ splits completely in $\mathbb{Q}(\zeta_p)$. There is a unique degree $p$-extension of $\mathbb{Q}(\zeta_p)$ contained in $\mathbb{Q}(\zeta_p, \zeta_\ell)$, call this extension $K$. Using the ambiguous class number formula it is possible to show that the $p$-torsion of the class group of $K$ is non-trivial.
How does the $p$-torsion of the class group of $K$ depend on $\ell$? What happens as we vary over all $\ell$? Is it possible that the $p$-torsion of the class group of $K$ becomes arbitrarily large or is there an upper bound (depending on $\ell$ maybe)?

REPLY [2 votes]: I do not think that it is possible to even bound the $l$-rank of ${\mathbb Q}(\zeta_p)$ in any simple way. Experiments with pari in the case $p = 5$ yield the following results for the $l$-rank $r$ of $K$:

$r = 1$ for $l = 11$, $41$, $71$, $191$, $241$, $271$, $\ldots$
$r  =2$ for $l = 31$, $131$, $151$, $251$, $\ldots$
$r = 3$ for $l = 101$, $181$, $281$, $\ldots$
$r = 4$ for $l = 211$, $401$, $\ldots$

If $l$ does not divide the class number of ${\mathbb Q}(\zeta_p)$, then the ambiguous class group has rank $\le p-1$, which implies that the $l$-rank is at most $(l-1)(p-1)$ (this is due to Inaba; see Prop. 2.4
in Chapter 2 here.