Local units and circular index in abelian \(p\)-group rings
The paper is concerned with units in integral group rings \(\mathbb{Z} A\) of abelian \(p\)-groups \(A\), \(p\neq 2\). First a \(\mathbb{Z}_ p\)-basis of the form \(w_ p(z)\), \(1\neq z\in A\) for \(\{1+ (z-1)+ (z^{-1}-1)\): \(z\in A\} \subset \mathbb{Z}_ p A\) is given, where \(w_ p(x)\) is a certain polynomial over \(\mathbb{Z}_ p\). Then a subgroup \(\Lambda\) in \((\mathbb{Z} A)^ \times\) of a similar structure is built which lies in the group \(\Omega\) of all units appearing circular under every character of \(A\). Finally the index \([\Omega: \Lambda]\) is shown to be the order of a \(D\)-group of ideal classes of \(\mathbb{Z} A\).
scientific article; zbMATH DE number 166146