On the cyclotomic unit group and the \(p\)-ideal class group of a real abelian number field
The setting of this paper is practically identical to the setting in two previous papers of the same author [\textit{M. Ozaki}, J. Number Theory 64, 211-222 (1997; Zbl 0879.11058) and ibid. 223-232 (1997; Zbl 0879.11059)]: \(K\) is a real abelian extension of the rationals, \(K_n\) the \(n\)-th layer in the \(p\)-cyclotomic extension (\(p\) a fixed prime), and one considers two Galois modules attached to \(K_n\): \(A_n\) is the \(p\)-part of the class group of \(K_n\), and \(B_n\) is the \(p\)-part of the quotient of the units of \(K_n\) modulo the cyclotomic units, in the sense of Sinnott. For some classes of fields \(K\), one can prove that \(A_n\) and \(B_n\) are isomorphic [see the paper just quoted, and also the following two papers: \textit{J. Kraft} and \textit{R. Schoof}, Compos. Math. 97, 135-155 (1995; Zbl 0840.11043); \textit{J. Kim, S. Bae} and \textit{J. Lee}, Isr. J. Math. 75, 161-165 (1991; Zbl 0765.11042)]. This time the author proves a weaker result for a slightly wider class of fields \(K\): a quotient of \(A_n\) embeds into \(B_n\).
scientific article; zbMATH DE number 1130805