A generalization of Maillet and Demyanenko determinants for the cyclotomic \(\mathbb Z_p\)-extension (Q5956159)

Let \(K\) be an imaginary Abelian field, \(h_K^-\) the relative class number of \(K\). In his previous paper [Acta Arith. 83, 391--397 (1998; Zbl 0895.11045)], the author gave a formula for \(h_K^-\) in the form of a determinant, which generalizes both formulae for Maillet and Demyanenko determinants. In the present paper, for an odd prime \(p\) and \(K\) as above, the author considers his result for the \(n\)-th layer \(K_n\) in the cyclotomic \(\mathbb Z_p\)-extension of \(K\) and decomposes his determinant into the product of \(p^n\) determinants, each of degree \([K:\mathbb Q]/2\), by means of characters of \(\text{Gal}(K_n/K)\).