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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.


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