Class groups in cyclic $\ell$-extensions: Comments on a paper by G. Cornell

Abstract:

Let $E/F$ be a cyclic extension of number fields of degree $\ell ^{n}$, where $\ell$ is a prime. It is proved that the $\ell$-rank of the class group of $E$ is bounded by $\ell ^{n}(t-1+ \mathrm {rk}_\ell Cl_{F})$, where $t$ is the number of primes of $F$, including infinite primes, which ramify in $E$ and $Cl_{F}$ is the class group of $F$. This generalizes a result of G. Cornell which applies when $n=1$ and $\ell$ is odd. A similar result is shown to hold when the Galois group is an abelian $\ell$-group and the Hasse norm theorem is valid for $E/F$.