zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 2 -extension of . (English) Zbl 1189.11033

Let Ω n be the field Ω n =(2cos(2π/2 n+2 )). Let h n denote the class number of Ω n . The author has proved the following theorem:

If is a prime number less than 10 7 , then for all n1, does not divide h n .

The previous results in this area were given by several authors, in particular 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 n1, L. C. Washington [Class Numbers and p -Extensions,” Math. Ann. 214, 177–193 (1975; Zbl 0302.12007)] who proved that, for a fixed prime , the -part of h n is bounded as n, and also K. Horie through several papers [”The Ideal Class Group of the Basic 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 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 be prime number:

1) If 3,5mod8 then l does not divide h n for all n1.

2) If 9mod16 and >34797970939, then does not divide h n for all n1.

3) If 7mod16 and l>210036365154018, then does not divide h n for all n1.

Using a Sinnott and Washington’s method, see [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 be an odd prime number and 2 c the exact power of 2 dividing -1 or 2 -1 according as 1mod4 or not. Let δ l denote 0 or 1 according as 1mod4 or not. Put m=3c-1+2[log 2 (-1)]-2δ l where, for a real x, [x] denotes the largest integer not exceeding x. If does not divide the class number of Ω m , then does not divide the class number of Ω n for all n1.

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