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Weber’s class number problem in the cyclotomic $\Bbb Z_2$-extension of $\Bbb Q$. (English) Zbl 1189.11033
Let $\Omega_n$ be the field $\Omega_n=\Bbb Q(2\cos(2\pi/2^{n+2}))$. Let $h_n$ denote the class number of $\Omega_n$. The author has proved the following theorem: If $\ell$ is a prime number less than $10^7$, then for all $n\geq 1$, $\ell$ does not divide $h_n$. The previous results in this area were given by several authors, in particular {\it H. Weber} [”Theorie der Abel’schen Zahlkörper,” Acta Math. 8, 193--263; ibid. 9, 105--130 (1886; JFM 18.0055.04)] who proved that $h_n$ is odd for all $n\geq 1$, {\it L. C. Washington} [Class Numbers and $\Bbb Z_p$-Extensions,” Math. Ann. 214, 177--193 (1975; Zbl 0302.12007)] who proved that, for a fixed prime $\ell$, the $\ell$-part of $h_n$ is bounded as $n\to\infty$, and also {\it K. Horie} through several papers [”The Ideal Class Group of the Basic $\Bbb Z_p$-Extension over an Imaginary Quadratic Field,” Tôhoku Math. J. (2) 57, No. 3, 375--394 (2005; Zbl 1128.11051), ”Triviality in Ideal Class Groups of Iwasawa-Theoretical Abelian Number Fields,” J. Math. Soc. Japan 57, No. 3, 827--857 (2005; Zbl 1160.11357), ”Primary Components of the Ideal Class Groups of Iwasawa-Theoretical Abelian Number Fields,” J. Math. Soc. Japan 59, No. 3, 811--824 (2007; Zbl 1128.11052), ”Certain Primary Components of the Ideal Class Group of the $\Bbb Z_p$-Extension over the Rationals,” Tôhoku Math. J. (2) 59, No. 2, 259--291 (2007; Zbl 1202.11050)]. In particular, a very effective result of Horie was: Let $\ell$ be prime number: 1) If $\ell\equiv 3,5\bmod 8$ then $l$ does not divide $h_n$ for all $n\geq 1$. 2) If $\ell\equiv 9\bmod 16$ and $\ell> 34797970939$, then $\ell$ does not divide $h_n$ for all $n\geq 1$. 3) If $\ell\equiv 7\bmod 16$ and $l>210036365154018$, then $\ell$ does not divide $h_n$ for all $n\geq 1$. Using a Sinnott and Washington’s method, see [{\it L. C. Washington}, Introduction to Cyclotomic Fields. 2nd ed. Graduate Texts in Mathematics. 83. New York, NY: Springer (1997; Zbl 0966.11047), section 16.3], the author proves the intermediate result: Let $\ell$ be an odd prime number and $2^c$ the exact power of $2$ dividing $\ell-1$ or $\ell^2-1$ according as $\ell\equiv 1\bmod 4$ or not. Let $\delta_l$ denote $0$ or $1$ according as $\ell\equiv 1\bmod 4$ or not. Put $m=3c-1+2[log_2(\ell-1)]-2\delta_l$ where, for a real $x$, $[x]$ denotes the largest integer not exceeding $x$. If $\ell$ does not divide the class number of $\Omega_m$, then $\ell$ does not divide the class number of $\Omega_n$ for all $n\geq 1$. His main theorem is then deduced from this result with some algebraic numbers numerical computations.

MSC:
11G15Complex multiplication and moduli of abelian varieties
11R18Cyclotomic extensions
11R27Units and factorization
11R29Class numbers, class groups, discriminants
11Y40Algebraic number theory computations
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