### 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 |
---|---|

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 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

_{2}-extension of ℚ, II.

*Journal de Theorie des Nombres de Bordeaux*,

*22*(2), 359-368. https://doi.org/10.5802/jtnb.720

**Weber’s class number problem in the cyclotomic ℤ _{2}-extension of ℚ, II.** / Fukuda, Takashi; Komatsu, Keiichi.

Research output: Contribution to journal › Article

_{2}-extension of ℚ, II',

*Journal de Theorie des Nombres de Bordeaux*, vol. 22, no. 2, pp. 359-368. https://doi.org/10.5802/jtnb.720

_{2}-extension of ℚ, II. Journal de Theorie des Nombres de Bordeaux. 2010;22(2):359-368. https://doi.org/10.5802/jtnb.720

}

TY - JOUR

T1 - Weber’s class number problem in the cyclotomic ℤ2-extension of ℚ, II

AU - Fukuda, Takashi

AU - Komatsu, Keiichi

PY - 2010

Y1 - 2010

N2 - Let hn denote the class number of n-th layer of the cyclotomic ℤ2-extension of ℚ. Weber proved that hn (n ≥ 1) is odd and Horie proved that hn (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ ≡ 3, 5 (mod 8). In a previous paper, the authors showed that hn (n ≥ 1) is not divisible by a prime number ℓ less than 107. In this paper, by investigating properties of a special unit more precisely, we show that hn (n ≥ 1) is not divisible by a prime number ℓ less than 1.2 • 108. Our argument also leads to the conclusion that hn (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ = ± 1 (mod 16).

AB - Let hn denote the class number of n-th layer of the cyclotomic ℤ2-extension of ℚ. Weber proved that hn (n ≥ 1) is odd and Horie proved that hn (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ ≡ 3, 5 (mod 8). In a previous paper, the authors showed that hn (n ≥ 1) is not divisible by a prime number ℓ less than 107. In this paper, by investigating properties of a special unit more precisely, we show that hn (n ≥ 1) is not divisible by a prime number ℓ less than 1.2 • 108. Our argument also leads to the conclusion that hn (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ = ± 1 (mod 16).

UR - http://www.scopus.com/inward/record.url?scp=85009959329&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009959329&partnerID=8YFLogxK

U2 - 10.5802/jtnb.720

DO - 10.5802/jtnb.720

M3 - Article

VL - 22

SP - 359

EP - 368

JO - Journal de Theorie des Nombres de Bordeaux

JF - Journal de Theorie des Nombres de Bordeaux

SN - 1246-7405

IS - 2

ER -