An index formula for the relative class number of an abelian number field

Abstract

Let n be the conductor of an imaginary abelian number field K, O the ring of algebraic integers of K, and Q n the nth cyclotomic field. We describe the index of the additive group generated by the conjugate elements of the trace Tr Q n / K ( i·cot(π/ n)) in the group O ⌣ i· R (if n = p r is a prime power, one has to take Tr Q n / K ( ip·cot(π/ n)) instead). This index equals the relative class number of K, multiplied with a factor that is explicitly given in terms of the ramification of K over Q . In some respect this result is an analogue of the representation of the class number of K ⌣ R as an index of circular units, rather than the hitherto known Stickelberger index formulas of Iwasawa et al. We also show that the higher derivatives i m cot (m − 1)( π n ) , m ≥ 2, yield index formulas analogous to the higher Stickelberger formulas of Kubert and Lang.