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A note on Sinnott’s index formula. (English) Zbl 0887.11046

Let \(K\) be an imaginary, abelian number field with conductor \(m= p_1^{e_1} p_2^{e_2}\). Supposing that the maximal subfields of \(K\) with conductor a prime power are also imaginary, K. Dohmae [J. Number Theory 61, 343-364 (1996; Zbl 0869.11082)] found a basis for the group of circular units of \(K\) and a simple, direct proof for Sinnott’s class number formula. Dropping the above supposition, the present paper yields analogous results. In particular, one obtains \(g'=\delta_p + \delta_q\), which makes these 3 parameters disappear in Sinnott’s index formula (compare Theorems 4.1 and 5.1 in [W. Sinnott, Invent. Math. 62, 181-234 (1980; Zbl 0465.12001)]).
Reviewer: G.Lettl (Graz)

MSC:

11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
11R20 Other abelian and metabelian extensions