Formulae for the relative class number of an imaginary abelian field in the form of a determinant

Abstract

Abstract. There is in the literature a lot of determinant formulae involving the relative class number of an imaginary abelian field. Usually such a formula contains a factor which is equal to zero for many fields and so it gives no information about the class number of these fields. The aim of this paper is to show a way of obtaining most of these formulae in a unique fashion, namely by means of the Stickelberger ideal. Moreover some new and non-vanishing formulae are derived by a modification of Ramachandra's construction of independent cyclotomic units. §1. Introduction

Determinant formulas for class numbers in function fields

“…Recently the authors gave determinant formulas for the real and relative class number of any subfield of a cyclotomic field with prime power conductor [2]. In this paper, by extending Kučera's idea [8,Lemma 2] to the function field case, we obtain several determinant formulas involving the real class number and the relative class number of any subfield of cyclotomic function fields with arbitrary conductor.…”

“…For these determinant formulas, we refer to Kučera's paper [8], where one can find the history of Maillet's determinant and Demjanenko matrix and many important results about them. In the same paper, Kučera showed a way of obtaining most of these determinant formulas in a unique fashion by means of the Stickelberger ideal.…”

Class numbers of some abelian extensions of rational function fields

“…In section 2, we give some notation for cyclotomic extensions and their subfields of rational function fields, and we also state some earlier results needed in this paper. In section 3, we extend Kučera's lemma [Ku,Lemma 2] to the function field case. By using this lemma, we obtain determinant class number formulas for h − (K) and h(K + ) (Theorem 3.2).…”

On a generalization of the Maillet determinant II

“…The non-vanishing condition of δ t 1 as well as the excepted cases k = 1, t = 1, 3 are rather delicate. Therefore, we shall consider them later elsewhere (see however [12], [17]). …”