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

There is in the literature a lot of determinant formulae involving the rela tive class number of an imaginary abelian field. Usually such a formula con tains a factor which is equal to zero for many fields and so it gives no in formation 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, name ly by means of the Stickelberger ideal. Moreover some new and non-vanishing formulae are derived by a modification of Ramachandra's construction of in dependent cyclotomic units.