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A class number problem in the cyclotomic \(\mathbb Z_3\)-extension of \(\mathbb Q\). (English) Zbl 1205.11116
Let \(\Omega_n=\mathbb Q\left(2\cos\bigl(\frac{2\pi}{3^{n+1}}\bigr)\right)\). This is the \(n\)-th lyer of the cyclotomic \(\mathbb Z_3\)-extension of \(\mathbb Q\). Let \(h_n\) be the class number of \(\Omega_n\). Let \(l\geq 5\) be a prime number and \(3^s\) the exact power of \(3\) dividing \(l^2-1\). Put \[ m_l=3s+2+[\log_3(l-1)]+\left[\log_3\frac{l-1}{2}\right]+[\log_3(2s+1+[\log_3(l-1)])], \] where \([x]\) denotes the greatest integer not exceeding \(x\). The author proves that if \(l\) does not divide \(h_{m_l}\), then \(l\) does not divide \(h_n\) for any positive integer \(n\). As a corollary, if \(l<10000\) then \(l\) does not divide \(h_n\) for any positive integer \(n\).

11R23 Iwasawa theory
11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions
Full Text: DOI
[1] T. Fukuda and K. Komatsu, Weber’s Class Number Problem in the Cyclotomic \(\Z_2\)-Extension of \(\Q\), to appear in Experiment., Math. · Zbl 1189.11033
[2] M. Aoki and T. Fukuda, An Algorithm for Computing \(p\)-Class Groups of Abelian Number Fields, Algorithmic Number Theory, 56-71,Lecture Notes in Computer Science, vol. 4076, Springer, Berlin, 2006. · Zbl 1143.11368
[3] K. Horie, Certain Primary Components of the Ideal Class Group of the \( \Z_p \)-Extension over the Rationals, Tohoku Math. J., 59 (2007), 259-291. · Zbl 1202.11050
[4] K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 257-258. · Zbl 0074.03002
[5] F. J. van der Linden, Class Number Computations of Real Abelian Number Fields, Math. Comp., 39 (1982), 693-707. JSTOR: · Zbl 0505.12010
[6] J. M. Masley, Class numbers of real cyclic number fields with small conductor, Compositio Math., 37 (1978), 297-319. · Zbl 0428.12003
[7] L. C. Washington, Introduction to Cyclotomic Fields , 2nd edition, Graduate Texts in Math. 83, Springer-Verlag, New York, Heidelberg, Berlin, 1997. · Zbl 0966.11047
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