Gauss units in integral group rings
In a number of papers the generation of the unit group \(U(RG)\) of an integral group ring \(RG\) of a finite group \(G\) over some ring \(R\) of algebraic integers has been studied, and often a finite set of explicit units could be given that generates \(U(RG)\) up to a finite index. In this paper \(R\) is the ring of integers in \(\mathbb{Q}(\sqrt{-p}:p\mid m)\) where \(m\in\mathbb{N}\) only allows prime divisors \(p\equiv 3\bmod 4\), but is itself not a power of 3, and where \(G\) is generated by \(a,b\) subject to the relations \(a^m=1=b^n\) \(a^b=a^{-1}\), \(\gcd(n,m)=1\), \(n\equiv 0\bmod 4\). Such \(G\) have fixed point free representations and so Bass units and bicyclic units will not suffice for getting a finite index in \(U(RG)\). The new units introduced here, which replace the bicyclic ones and guarantee a finite index, are so-called Gauß units and look like \(1+x^+yx^-\), \(1+x^-yx^+\) with \(y\in RG\) and \(x^\pm=(\widehat g-p^k)\cdot\sum_{j=0}^{p^k-1}\sqrt{-p} g^{j^2}\pm g^{p^{j^2}}\) for \(g\in G\) of order \(p^k\).
scientific article; zbMATH DE number 1218676