On the index of the group generated by relative units
For an algebraic number field \(K\), denote by \(E_ K\) and \(W_ K\) the group of units and the group of roots of unity in \(K\), respectively. Consider a fixed abelian extension \(L\) of degree \(n\) over \(K\), with \(G= \text{Gal} (L/K)\). Let \(M\) be an intermediate field which is cyclic over \(K\). This article deals with the relative units of \(M/K\), i.e., the elements \(\varepsilon\in E_ M\) for which the norm \(N_{M/F} \varepsilon\) belongs to \(W_ F\) whenever \(F\) is an extension of \(K\) properly contained in \(M\). Let \(E_{M/K}\) denote the group of these relative units and write \({\mathcal E}= \prod_ M E_{M/K} W_ L/W_ L\), the product being over all possible \(M\). The author finds upper bounds for the index \([E:{\mathcal E}]\), where \(E= E_ L/ W_ L\). His previous results [Proc. Jap. Acad., Ser. A 64, 304-306 (1988; Zbl 0672.12002)] showed that this index divides \(n^ r\), where \(r\) is the \(\mathbb{Z}\)-rank of \(E\). Now he introduces a group \({\mathbf E}\) with \({\mathcal E}\leq {\mathbf E} \leq E\) and majorizes the indices \([{\mathbf E}:{\mathcal E}]\) and \([E:{\mathbf E}]\) separately. The group \({\mathbf E}\) depends on the decomposition of the \(\mathbb{Q} [G]\)-module \(E\otimes_ \mathbb{Z} \mathbb{Q}\) through the \(\mathbb{Q}\)- irreducible characters of \(G\); this allows one to reduce the study to the individual components thus obtained.
scientific article; zbMATH DE number 556372