On the unit group of the group ring \(\mathbb{Z}[G]\). (Q1812490)

Exploiting work of \textit{T. Sekiguchi} and \textit{N. Suwa} on group schemes [Compos. Math. 97, No. 1-2, 253-271 (1995; Zbl 0876.14031)] the author observes that if \(G\) is a cyclic group of prime power order \(p^n\), then \((\mathbb{Z} G)^\times\simeq\{\pm 1\}\times\prod^n_{i=1}U^{{n\choose i}}_i\), where \(U_i=\{\varepsilon\in(\mathbb{Z}[\zeta_p]^{\otimes i})^\times:\varepsilon\equiv 1^{\otimes i}\bmod{(\zeta_p-1)}^{\otimes i}\}\). He then, for \(G=\mathbb{Z}/p\times\mathbb{Z}/p\), constructs units \(u_1,\dots,u_r\) in \(\mathbb{Z} G\) so that each \(u\in(\mathbb{Z} G)^\times\) is uniquely represented in the form \(\pm gu^{n_1}_1\cdots u^{n_r}_r\) with \(g\in G\) and \(n_i\in\mathbb{Z}\). Concrete examples for \(p=5,\,7\) are added. -- This is different, and does not follow, from Hoechsmann's work on units of \(\mathbb{Z} G\) for Abelian \(G\).