Sands, Jonathan W.; Schwarz, Wolfgang A Demjanenko matrix for abelian fields of prime power conductor. (English) Zbl 0829.11054 J. Number Theory 52, No. 1, 85-97 (1995). 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 \(h^- (K)\). Reviewer: V.Ennola (Turku) Cited in 2 ReviewsCited in 8 Documents MSC: 11R20 Other abelian and metabelian extensions 11R29 Class numbers, class groups, discriminants Keywords:Abelian field of prime power conductor; Demyanenko matrix; relative class number; upper bound PDFBibTeX XMLCite \textit{J. W. Sands} and \textit{W. Schwarz}, J. Number Theory 52, No. 1, 85--97 (1995; Zbl 0829.11054) Full Text: DOI