A Demjanenko matrix for abelian fields of prime power conductor
Fix a power \(q = p^r\) of an odd prime \(p\), and write \(G = (\mathbb{Z}/q \mathbb{Z})^*\), \(M = \{\overline a \in G : 0 < a \leq (q - 1)/2\}\). Let \(c_M\) be the characteristic function of \(M\). The corresponding Demyanenko matrix is \(D_q = (c_M (\overline a \overline b))_{\overline a, \overline b \in M}\). The determinant of \(D_q\) provides a formula for the relative class number of \(\mathbb{Q} (\zeta_q)\). The goal of the paper is to extend the theory to allow consideration of a quotient group of \(G\) and the corresponding subfield \(K\) of \(\mathbb{Q} (\zeta_q)\). The connection between the Demyanenko matrix and the real cyclotomic units leads to a simple proof of a parity result due to Hasse and Garbanati. The authors establish an expression for the relative class number \(h^- (K)\) as the determinant of a matrix with relatively small integer entries. This leads to a reasonable upper bound for \(h^- (K)\).
scientific article; zbMATH DE number 765163