### Abstract

Let h_{n} denote the class number of n-th layer of the cyclotomic ℤ_{2}-extension of ℚ. Weber proved that h_{n} (n ≥ 1) is odd and Horie proved that h_{n} (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ ≡ 3, 5 (mod 8). In a previous paper, the authors showed that h_{n} (n ≥ 1) is not divisible by a prime number ℓ less than 10^{7}. In this paper, by investigating properties of a special unit more precisely, we show that h_{n} (n ≥ 1) is not divisible by a prime number ℓ less than 1.2 • 10^{8}. Our argument also leads to the conclusion that h_{n} (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ = ± 1 (mod 16).

Original language | English |
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Pages (from-to) | 359-368 |

Number of pages | 10 |

Journal | Journal de Theorie des Nombres de Bordeaux |

Volume | 22 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 |

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Fukuda, T., & Komatsu, K. (2010). Weber’s class number problem in the cyclotomic ℤ

_{2}-extension of ℚ, II.*Journal de Theorie des Nombres de Bordeaux*,*22*(2), 359-368. https://doi.org/10.5802/jtnb.720