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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 $$n$$.

##### MSC:
 11R23 Iwasawa theory 11R29 Class numbers, class groups, discriminants 11R18 Cyclotomic extensions
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##### References:
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