### Abstract

In the preceding papers, two of authors developed criteria for Greenberg conjecture of the cyclotomic Z_{2}-extension of k = Q(√p ) with prime number p. Criteria and numerical algorithm in [5], [3] and [6] enable us to show λ2(k) = 0 for all p less than 105 except p =13841; 67073. All the known criteria at present can not handle p = 13841; 67073. In this paper, we develop another criterion for λ2(k) = 0 using cyclotomic units and Iwasawa polynomials, which is considered a slight modification of the method of Ichimura and Sumida. Our new criterion fits the numerical examination and quickly shows that λ2(Q(√p )) = 0 for p = 13841; 67073. So we announce here that λ2(Q(√p)) = 0 for all prime numbers p less that 10^{5}.

Original language | English |
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Pages (from-to) | 7-17 |

Number of pages | 11 |

Journal | Functiones et Approximatio, Commentarii Mathematici |

Volume | 54 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 |

### Keywords

- Cyclotomic unit
- Iwasawa invariant
- Real quadratic field

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

_{2}-extension of Q(√p), III.

*Functiones et Approximatio, Commentarii Mathematici*,

*54*(1), 7-17. https://doi.org/10.7169/facm/2016.54.1.1