Resumen de ℓ-class groups of fields in Kummer towers

Jianing Li, Yi Ouyang, Yue Xu, Shenxing Zhang

• Let ℓ and b be prime numbers and Kn,m = Q(p1 n , ζ2`m). We study the `-class group of Kn,m in this paper. When ` = 2, we determine the structure of the 2-class group of Kn,m for all (n, m) ∈ Z 2 ≥0 in the case p ≡ 3, 5 mod 8, and for (n, m) = (n, 0), (n, 1), or (1, m) in the case p ≡ 7 mod 16, generalizing the results of Parry about the 2-divisibility of the class number of K2,0. We also obtain results about the `-class group of Kn,m when ` is odd and in particular when ` = 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 `-class groups in the 2-dimensional Kummer tower {Kn,m}.