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The ideal class group of the \(Z_{23}\)-extension over the rational field. (English) Zbl 1234.11146

For an odd prime \(p\), let \({\mathbb B}_\infty\) denote the \({\mathbb Z}_p\)-extension of the rational field. In several papers the first author and later both authors jointly have studied the \(l\)-class group of \({\mathbb B}_\infty\) for primes \(l\) which are primitive roots mod \(p^2\). After the last paper in this series, published in [Abh. Math. Semin. Univ. Hamb. 80, No. 1, 47–57 (2010; Zbl 1214.11125)], we know that this group is trivial for \(p \leq 19\). In the present article the authors continue this study and obtain the same result for \(p=23\). They use the method of their earlier work, combining an arithmetic study of the analytic class number formula with numerical calculations performed by a personal computer.

MSC:

11R23 Iwasawa theory
11R20 Other abelian and metabelian extensions
11R29 Class numbers, class groups, discriminants
11Y40 Algebraic number theory computations

Citations:

Zbl 1214.11125
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References:

[1] K. Horie, Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field, J. London Math. Soc. (2) 66 (2002), no. 2, 257-275. · Zbl 1011.11072 · doi:10.1112/S0024610702003502
[2] K. Horie, Primary components of the ideal class group of the \(\mathbf Z_ p\)-extension over \(\mathbf Q\) for typical inert primes, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 3, 40-43. · Zbl 1114.11086 · doi:10.3792/pjaa.81.40
[3] K. Horie and M. Horie, The narrow class groups of some \(\mathbb Z_ p\)-extensions over the rationals, Acta Arith. 135 (2008), no. 2, 159-180. · Zbl 1158.11046 · doi:10.4064/aa135-2-5
[4] K. Horie and M. Horie, The narrow class groups of the \(\mathbb Z_{17}\)- and \(\mathbb Z_{19}\)-extensions over the rational field, Abh. Math. Sem. Univ. Hamburg. (to appear). · Zbl 1214.11125
[5] K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257-258. · Zbl 0074.03002
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