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A class number formula for the \(p\)-cyclotomic field. (English) Zbl 1120.11043

Let \(p\) be an odd prime number and \(G=\text{Gal}({\mathbb Q} (\zeta _p)/{\mathbb Q})\) the Galois group of \(p\)-th cyclotomic field \({\mathbb Q} (\zeta _p)\) over \({\mathbb Q}\). Let \(H\) be a subgroup of \(G\). For an element \(\alpha \in {\mathbb Z}[G]\), let \(\alpha_H=\sum_{\sigma \in H} a_{\sigma}\sigma\) denote the \(H\)-part of \(\alpha\). Then the author defined in [J. Math. Soc. Japan 58, 885–902 (2006; Zbl 1102.11059)] a Stickelberger ideal \({\mathcal{S}}_H\) of the group ring \({\mathbb Z}[H]\) by \({\mathcal{S}}_H = \{ \alpha_H \mid \alpha \in {\mathcal{S}}_G \} \in H\), where \({\mathcal S}_G\) denotes the ordinary Stickelberger ideal of \(\mathbb Z[G]\).
This paper deals with the relation between the relative class number \(h_p^{-}\) of \({\mathbb Q} (\zeta _p)\) and index of \({\mathcal{S}}_H\) in \(H\). Let \(p\) be a prime number with \(p \equiv 3 \pmod{4}\), and let \(H\) denote the subgroup of \(G\) with \([G:H]=2\). Then the main theorem of this paper states \(h_p^{-}/h({\mathbb Q}(\sqrt{-p}))=[{\mathbb Z}[H]:{\mathcal{S}}_H]\); this result refines in a sense the classical Iwasawa formula \(h_p^{-1}=[{\mathbb Z}[G]^{-}:{\mathcal S}_G^{-}]\)

MSC:

11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants

Citations:

Zbl 1102.11059
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