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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:
11R23Iwasawa theory
11R29Class numbers, class groups, discriminants
11R18Cyclotomic extensions
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References:
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[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