Construction of bases for the group of cyclotomic units
Let \(\varepsilon_n\) be a primitive \(n\)th root of unity and \(D^{(n)}\) the multiplicative group generated by \(1-\varepsilon_n^k\) with \(k\not\equiv 0\bmod n\) modulo roots of unity. The group of cyclotomic units \(C^{(n)}\) is the subgroup of \(D^{(n)}\) consisting of the elements which are units in \(\mathbb{Z}[\varepsilon_n]\). Previously, \textit{R. Kučera} [J. Number Theory 40, 284-316 (1992; Zbl 0744.11052)] has constructed a basis for \(C^{(n)}\). Here the author introduces a new interesting approach which leads to a basis for \(C^{(n)}\) and their union \(C^{(\infty)}.\) The starting point is a finitely generated free \(\mathbb{Z}\)-module \(M\) with an involution \(\sigma\) operating on it. The author defines the notion of a weak \(\sigma\)-basis of \(M\) which readily induces bases of \(M_+=M/\ker (1+\sigma)\) and \(M_-=M/\ker (1-\sigma)\). The next step is to consider a family of triplets \((M_d,\mathcal E_d,\text{n}_d)_{d\in\Delta}\), where \(\Delta\) is an ordered index set, the \(M_d\) are modules, \(\mathcal E_d\subseteq M_d\), and \(\text{n}_d\) maps \(\mathcal E_d\) into the direct sum of \(M_t\) with \(t<d\). The combination \(\mathcal L\) of the system is the quotient module of the direct sum of \(M_d\) over the submodule generated by the sums \(r+\text{n}_d(r)\) with \(r\in\mathcal E_d\). The weak \(\sigma\)-bases of \(M_d/\langle\mathcal E_d\rangle\) then allow one to construct a basis of \(\mathcal L_+\). It is possible to choose the system so that for the combination \(\mathcal L=\mathcal L(n)\) we have \(\mathcal L(n)_+\simeq D^{(n)}\). This leads to a basis of \(D^{(n)}\) which can be modified to a basis of \(C^{(n)}\). In the last section there is a short discussion on further applications of the method. These are the explicit construction of relations in \(C^{(n)}\) and similar results for the Stickelberger ideal as well as for cyclotomic units.
scientific article; zbMATH DE number 1420261
scientific article