On Demjanenko's matrix and Maillet's determinant for imaginary abelian number fields (Q1924212)

\textit{J. Sands} and \textit{W. Schwarz} [J. Number Theory 52, 85-97 (1995; Zbl 0829.11054); cf. also the review of \textit{K. Dohmae}, Proc. Japan Acad., Ser. A 70, 292-294 (1994; Zbl 0829.11055)] defined a generalization of Demjanenko's matrix for an imaginary abelian number field \(K\) of odd prime power conductor. The author defines a further generalization \(\Delta(K,l)\), where \(K\) is any imaginary abelian field and the parameter \(l\) is an integer prime to the conductor of \(K\). The author computes the determinant \(\text{det } \Delta(K,l)\) and shows that in the case \(l=n+1\) it coincides with Maillet's determinant generalized by \textit{K. Girstmair} [Math. Comput. 61, 881-887 (1993; Zbl 0787.11046)]. As an application, an upper bound for the relative class number \(h^-(K)\) is given in the case when the conductor of \(K\) is a power of 2.