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A class number problem in the cyclotomic $\Bbb Z_3$-extension of $\Bbb Q$. (English) Zbl 1205.11116
Let $\Omega_n=\Bbb Q\left(2\cos\bigl(\frac{2\pi}{3^{n+1}}\bigr)\right)$. This is the $n$-th lyer of the cyclotomic $\Bbb Z_3$-extension of $\Bbb Q$. 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 $n$.

##### MSC:
 11R23 Iwasawa theory 11R29 Class numbers, class groups, discriminants 11R18 Cyclotomic extensions
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##### References:
 [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 · doi:10.1007/11792086 [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 · doi:10.2748/tmj/1182180736 [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 · doi:10.2307/2007347 · http://links.jstor.org/sici?sici=0025-5718%28198210%2939%3A160%3C693%3ACNCORA%3E2.0.CO%3B2-B&origin=euclid [6] J. M. Masley, Class numbers of real cyclic number fields with small conductor, Compositio Math., 37 (1978), 297-319. · Zbl 0428.12003 · numdam:CM_1978__37_3_297_0 · eudml:89385 [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