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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.

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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