The narrow class groups of some Z<sub>p</sub>-extensions over the rationals

Horie, Kuniaki; Horie, Mitsuko

doi:10.4064/aa135-2-5

Cited by 13 publications

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“…Modifying part of the proof of the above result, we have shown in [4,6] that, when p ≤ 13, the result holds without the second condition on l. Furthermore, in [6], we have proved the triviality of the narrow 2-class group of B ∞ for the case p ≤ 13, still by means of the analytic class number formula. It should be added that most of the computations of [6] need a personal computer, together with Mathematica. Meanwhile, in [5], the triviality of the l-class group of B ∞ is discussed for the case where l ≡ r 2 (mod p 2 ) with some primitive root r modulo p 2 .…”

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confidence: 78%

“…A result of Armitage and Fröhlich [1] on the signatures of units related to the rank of a 2-class group will then be applied to any subfield of B ∞ . As a consequence, we shall see from results of [5,6] that, when p = 17 or p = 19, the narrow 2-class group of B ∞ is trivial. Most of our computations are done with Mathematica as in [6].…”

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confidence: 91%

“…In this paper, we shall first prove by arguments similar to those of [4,6] that the l-class group of B ∞ is trivial if p = 17 or p = 19 and if l is a primitive root modulo p 2 . A result of Armitage and Fröhlich [1] on the signatures of units related to the rank of a 2-class group will then be applied to any subfield of B ∞ .…”

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confidence: 98%

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The narrow class groups of the ℤ17- and ℤ19-extensions over the rational field

Horie

2009

Abh. Math. Semin. Univ. Hambg.

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Let p be either 17 or 19, let Z p denote the ring of p-adic integers, and let l be a prime number which is a primitive root modulo p 2 . We shall prove, with the help of a computer, that the l-class group of the Z p -extension over the rational field is trivial. We shall also prove the triviality of the narrow 2-class group of the same Z p -extension. IntroductionLet p be an odd prime number. Let Z p denote the ring of p-adic integers, and B ∞ the Z p -extension over the rational number field Q, i.e., the unique abelian extension over Q, in the complex number field, whose Galois group over Q is topologically isomorphic to the additive group of Z p . For each nonzero element θ of B ∞ , let (θ ) denote the principal ideal of B ∞ generated by θ . We denote by J the ideal group of B ∞ , and by P the subgroup of J consisting of (θ ) for all totally positive elements θ of B ∞ . The quotient group J/P and the 2-primary component of J/P will be called the narrow class group of B ∞ and the narrow 2-class group of B ∞ , respectively. Obviously, for any odd prime q, the natural homomorphism of the narrow class group of B ∞ onto the ideal class group of B ∞ induces an isomorphism of the q-primary component of the narrow class group of B ∞ onto the q-class group of B ∞ . Communicated by U. Kühn.

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confidence: 78%

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confidence: 91%

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The narrow class groups of the ℤ17- and ℤ19-extensions over the rational field

Horie

2009

Abh. Math. Semin. Univ. Hambg.

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“…On the indivisibility problem, Washington [26] also showed that the minus part of the class number of Q(µ 5 n+1 ) is indivisible by every prime number with 8 ≡ 1 (mod 100) for every non-negative integer n. Later, K. Horie [7,8,9,10] and K. Horie-M. Horie [11,12,13,14,15] made a breakthrough. Indeed, they succeeded in controlling cyclotomic units which relate to our class numbers.…”

Section: Problem 11 Fix a Prime Number Is The Class Number H N Inmentioning

confidence: 99%

“…Later, several authors showed h n = 1 for (p, n) = (2, 4), (2,5), (3,1), (3,2), (3,3), (5,1) and (7, 1) (see [1], [3], [18] and [19]). And recently, J. C. Miller obtained striking results determining h n = 1 for (p, n) = (2, 6), (5,2), (11,1), (13,1), (17,1) and (19,1) (see [20] and [21]). However, calculating one class number by one gives information on the class numbers for only finitely many layers.…”

Section: Introductionmentioning

confidence: 99%