An index formula for the relative class number of an abelian number field
It is shown that the relative class number \(h^-_ K\), of an imaginary abelian field K (of \(conductor\quad n\) and ring of \(integers\quad {\mathfrak O}),\) is given by: \((*)\quad c_ Kh^-_ K=((1-\tau){\mathfrak O}:<\eta >),\) where \(<\eta >\) is the additive \({\mathbb{Z}}[Gal(K/{\mathbb{Q}})]\)-module generated by the trace \(\eta\), in \({\mathbb{Q}}(\zeta_ n)/K\), of \(i\cdot \cot g(\pi /n)=(1+\zeta_ n)/1-\zeta_ n)\) \((\zeta_ n=\exp (2i\pi /n)\), \(\tau =complex\) conjugation); here the rational \(c_ K\neq 0\) depends on standard Euler factors and elementary invariants of \(K/{\mathbb{Q}}.\) The computation of \(((1-\tau){\mathfrak O}:<\eta>)\) involves Leopoldt's ``character coordinates'' [\textit{H. W. Leopoldt}, J. Reine Angew. Math. 201, 119-149 (1959; Zbl 0098.034)] for odd character \(\chi\); for \(\eta\), these coordinates are essentially Bernoulli numbers \(B_{1,\chi}\), and the result follows from a classical analytic formula of \(h^-_ K\) (Kubert-Lang formulas are also described with similar indices).
scientific article; zbMATH DE number 4105076