Classnumbers and unit signatures

Armitage, J. V.; Fröhlich, A.

doi:10.1112/s0025579300008044

Cited by 42 publications

(60 citation statements)

References 1 publication

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“…Further B u /Q is a cyclic extension of degree p u . Therefore, in case the order of 2 modulo p is even, the assertion IV of [1] implies that every principal ideal of B u is generated by a totally positive element of B u , whence the narrow 2-class group of B u is trivial. Then follows the triviality of the narrow 2-class group of B ∞ .…”

Section: Lemma 3 (Cf [6 Lemma 2]) If There Exist Integers C and D Smentioning

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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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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Section: Lemma 3 (Cf [6 Lemma 2]) If There Exist Integers C and D Smentioning

confidence: 91%

“…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.…”

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confidence: 99%

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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“…This proof of the theorem of Armitage & Fröhlich [1] is essentially due to Oriat [23]. Proofs dual to Oriat's were given by Hayes [10], who argued using the Galois groups of the Kummer extensions corresponding to elements in Sel(F ).…”

Section: Associated Unit Groupsmentioning

confidence: 99%

Selmer groups and quadratic reciprocity

Lemmermeyer

2006

Abh.Math.Semin.Univ.Hambg.

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Abstract. In this article we study the 2-Selmer groups of number fields F as well as some related groups, and present connections to the quadratic reciprocity law in F .Let F be a number field; elements in F × that are ideal squares were called singular numbers in the classical literature. They were studied in connection with explicit reciprocity laws, the construction of class fields, or the solution of embedding problems by mathematicians like Kummer, Hilbert, Furtwängler, Hecke, Takagi, Shafarevich and many others. Recently, the groups of singular numbers in F were christened Selmer groups by H. Cohen [4] because of an analogy with the Selmer groups in the theory of elliptic curves (look at the exact sequence (2.2) and recall that, under the analogy between number fields and elliptic curves, units correspond to rational points, and class groups to Tate-Shafarevich groups).In this article we will present the theory of 2-Selmer groups in modern language, and give direct proofs based on class field theory. Most of the results given here can be found in § § 61ff of Hecke's book [11]; they had been obtained by Hilbert and Furtwängler in the roundabout way typical for early class field theory, and were used for proving explicit reciprocity laws. Hecke, on the other hand, first proved (a large part of) the quadratic reciprocity law in number fields using his generalized Gauss sums (see [3] and [22]), and then derived the existence of quadratic class fields (which essentially is just the calculation of the order of a certain Selmer group) from the reciprocity law.In Takagi's class field theory, Selmer groups were moved to the back bench and only resurfaced in his proof of the reciprocity law. Once Artin had found his general reciprocity law, Selmer groups were history, and it seems that there is no coherent account of their theory based on modern class field theory.Hecke's book [11] is hailed as a classic, and it deserves the praise. Its main claim to fame should actually have been Chapter VIII on the quadratic reciprocity law in number field, where he uses Gauss sums to prove the reciprocity law, then derives the existence of 2-class fields, and finally proves his famous theorem that the ideal class of the discriminant of an extension is always a square. Unfortunately, this chapter is not exactly bedtime reading, so in addition to presenting Hecke's results in a modern language I will also give exact references to the corresponding theorems in Hecke's book [11] in the hope of making this chapter more accessible.The actual reason for writing this article, however, was that the results on Selmer groups presented here will be needed for computing the separant class group of F , a new invariant that will be discussed thoroughly in [21].

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“…Les idempotents ~e^ i = 1,2,..., g, peuvent se calculer facilement et ils possèdent la propriété suivante qui nous sera utile ( [2] P.97). C'est une simple traduction des définitions et de la proposition 15.…”

Section: Propriétés De L'algèbre Ol = ¥^[G]/(v)unclassified

Signature des unités cyclotomiques et parité du nombre de classes des extensions cycliques de ${\bf Q}$ de degré premier impair

Gras¹,

Gras²

1975

Annales De L’institut Fourier

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Classnumbers and unit signatures

Cited by 42 publications

References 1 publication

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

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

Selmer groups and quadratic reciprocity

Signature des unités cyclotomiques et parité du nombre de classes des extensions cycliques de ${\bf Q}$ de degré premier impair

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