$\ell$-Class groups of fields in Kummer towers

Abstract

Let $\ell$ and $p$ be prime numbers and $K_{n,m}=\mathbb{Q}(p^{\frac{1}{{\ell }^n}},{\zeta }_{2{\ell }^m})$. We study the $\ell$-class group of $K_{n,m}$ in this paper. When $\ell =2$, we determine the structure of the $2$-class group of $K_{n,m}$ for all $(n,m)\in {\mathbb{Z}}_{\ge 0}^2$ in the case $p=2$ or $p\equiv 3,5\mathrm{mod}8$, and for $(n,m)=(n,0)$, $(n,1)$ or $(1,m)$ in the case $p\equiv 7\mathrm{mod}16$, eneralizing the results of Parry about the $2$-divisibility of the class number of $K_{2,0}$. We also obtain results about the $\ell$-class group of $K_{n,m}$ when $\ell$ is odd and in particular $\ell =3$. The main tools we use are class field theory, including Chevalley's ambiguous class number formula and its generalization by Gras, and a stationary result about the $\ell$-class groups in the $2$-dimensional Kummer tower $\{K_{n,m}\}$.

add the odd prime case. 22 pages

References in corpus (3)

• Invariant generalized ideal classes -- structure theorems for p-class groups in p-extensions
• A result of Lemmermeyer on class numbers
• On a Conjecture of Lemmermeyer

Cited by in corpus (5)

• On the $2$-class group of some number fields with large degree
• The rank of the 2-class group of some fields with large degree
• On Class Numbers of Pure Quartic fields
• The $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and its normal closure
• The Chevalley-Gras formula over global fields