Demjanenko matrix and 2-divisibility of class numbers
Let \(p\) be an odd prime and put \(g= (p-1)/2\), \(S= \{1,2,\dots, g\}\). The Demjanenko matrix \(H= (h_{ij})\) is the \(g\) by \(g\) matrix defined by \(h_{ij}= 1\) if \(i\cdot j\) belongs to \(S\bmod p\), and \(h_{ij}=0\) otherwise. Let \(E\) (resp. \(E_{\text{cyc}}\)) denote the group of units (resp. cyclotomic units) in the maximal real subfield of the \(p\)th cyclotomic field \(\mathbb{Q}(\zeta_ p)\), and let \(E^ +\), \(E_{\text{cyc}}^ +\) denote the subgroups of totally positive units. Finally, let \(h_ p^ +\) and \(h_ p^ -\) denote the plus and minus parts of the class number of \(\mathbb{Q}(\zeta_ p)\). \textit{F. Hazama} [J. Number Theory 34, 174-177 (1990; Zbl 0697.12003)] proved a formula which connects \(\text{det } H\) and \(h_ p^ -\). In a letter addressed to H. G. Zimmer on July 28, 1987, \textit{E. Reyssat} has discovered a connection between \(H\) and \(E_{\text{cyc}}\). Combining the two results, the author proves that \[ 2\mid h_ p^ - \iff E_{\text{cyc}}^ +\neq E_{\text{cyc}}^ 2 \iff 2\mid h_ p^ + \quad \text{or} \quad E^ +\neq E^ 2. \]
scientific article; zbMATH DE number 572583