# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 11G15 Complex multiplication and moduli of abelian varieties 11R18 Cyclotomic extensions 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants 11Y40 Algebraic number theory computations
Full Text: