Let be the field . Let denote the class number of . The author has proved the following theorem:
If is a prime number less than , then for all , does not divide .
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 is odd for all , L. C. Washington [Class Numbers and -Extensions,” Math. Ann. 214, 177–193 (1975; Zbl 0302.12007)] who proved that, for a fixed prime , the -part of is bounded as , and also K. Horie through several papers [”The Ideal Class Group of the Basic -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 -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 then does not divide for all .
2) If and , then does not divide for all .
3) If and , then does not divide for all .
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 the exact power of 2 dividing or according as or not. Let denote 0 or 1 according as or not. Put where, for a real , denotes the largest integer not exceeding . If does not divide the class number of , then does not divide the class number of for all .
His main theorem is then deduced from this result with some algebraic numbers numerical computations.