l-CLASS GROUPS OF FIELDS IN KUMMER TOWERS

摘要

Let l and p be prime numbers and K-n,K-m = Q(p(1/pln), sigma(2lm)). We study the l-class group of K-n,K-m in this paper. When l = 2, we determine the structure of the 2-class group of K-n,K-m for all (n,m) is an element of Z(>= 0)(2) in the case p equivalent to 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 K-2,K-0. We also obtain results about the l-class group of K-n,K-m when l is odd and in particular when l = 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 l-class groups in the 2-dimensional Kummer tower fKn;mg.

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