On the group of units of an abelian extension of an algebraic number field (Q1119967)

Let L/K be a finite abelian extension over a finite extension K of the rational number field \({\mathbb{Q}}\). For any cyclic sub-extension M of L/K, put \({\mathcal E}_ M=E_{M/K}W_ L/W_ L\), where \(E_{M/K}\) and \(W_ L\) mean the group of relative units of M over K and the group of roots of unity in L respectively. The author considers the structure of the group \({\mathcal E}_ M\), and generalizes results of \textit{G. Gras} and \textit{M.-N. Gras} [Bull. Sci. Math., II. Sér. 101, 97-129 (1977; Zbl 0359.12007)], \textit{H. W. Leopoldt} [Abh. Deutsch. Akad. Wiss., Math.-Naturwiss. Kl. 1953, No.2, 1- 48 (1954; Zbl 0059.035)] in the case \(K={\mathbb{Q}}\) and results of \textit{R. Schertz} [J. Reine Angew. Math. 295, 151-168 (1977; Zbl 0356.12009)], \textit{K. Nakamula} [J. Math. Soc. Japan 37, 245-273 (1985; Zbl 0591.12010)] in the case K is an imaginary quadratic field.