A generalization of Maillet and Demyanenko determinants

Abstract

where the product ∏ χ is taken over the odd primitive characters χ of K with conductor f(χ) (see Hasse [4]). Several ways of representing the product in (1) by a determinant are known, some of them holding for certain types of fields only (see Carlitz and Olson [1], Hazama [5] and further references in Hirabayashi [6]). Here our main concern is a unified approach to the Maillet determinant on the one hand and to the Demyanenko determinant on the other hand. This approach relies on the “b-division vector” introduced by Girstmair [3]. We obtain a relative class number formula for an arbitrary imaginary abelian number field which generalizes formulae of Girstmair [2] and [6]. Tsumura [7] also generalized both type of determinant formulae, but his generalization for the Demyanenko determinant requires the oddness of the conductor.