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2022 $\ell$-class groups of fields in Kummer towers
Jianing Li, Yi Ouyang, Yue Xu, Shenxing Zhang
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Publ. Mat. 66(1): 235-267 (2022). DOI: 10.5565/PUBLMAT6612210

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 $(n,m)\in \mathbb Z_{\geq 0}^2$ in the case $p\equiv 3, 5\bmod{8}$, and for $(n,m)=(n,0)$, $(n,1)$, or $(1,m)$ in the case $p\equiv 7\bmod{16}$, generalizing 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 when $\ell=3$. 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 $\{K_{n,m}\}$.

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Jianing Li. Yi Ouyang. Yue Xu. Shenxing Zhang. "$\ell$-class groups of fields in Kummer towers." Publ. Mat. 66 (1) 235 - 267, 2022. https://doi.org/10.5565/PUBLMAT6612210

Information

Received: 21 April 2020; Accepted: 15 September 2020; Published: 2022
First available in Project Euclid: 4 January 2022

Digital Object Identifier: 10.5565/PUBLMAT6612210

Subjects:
Primary: 11R11 , 11R16 , 11R18 , 11R20 , 11R29

Keywords: ambiguous class number formula , class group , Kummer tower

Rights: Copyright © 2022 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.66 • No. 1 • 2022
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