Class group of the group ring of a cyclic group of order \(p^ 2\) over the ring of integers in a quadratic number field
Let \({\mathfrak O}\Gamma_ n\) be the group ring of a cyclic group \(\Gamma_ n\) of order \(p^{n+1}\) (p an odd prime) over the ring of integers \({\mathfrak O}\) in a quadratic number field k (unramified at p), cl(\({\mathfrak O}\Gamma_ n)\) the locally free class group of \(\Gamma_ n\), cl(m) the class group of the maximal order m. It is well known that there exists the surjective homomorphism cl(\({\mathfrak O}\Gamma_ n)\to cl(m)\). The author investigates the kernel D(\({\mathfrak O}\Gamma_ n)\) of this homomorphism for \(n=1\). He shows that \(| D({\mathfrak O}\Gamma_ 1)|\) has a large p part which is minimal when p does not divide the class number h of \(k(\xi_ 0)\) where \(\xi_ 0\) is a primitive p-th root of unity. In particular the author proves that if \((p,h)=1\) then \(| D({\mathfrak O}\Gamma_ 1)| =\Omega | D({\mathfrak O}\Gamma_ 0)|\) where for k real \(p^{p-2}\) divides \(\Omega\) and for k imaginary \(p^{p-3}\) divides \(\Omega\).
scientific article; zbMATH DE number 3948418