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Note à propos d'une conjecture de H.J. Godwin sur les unités des corps cubiques

Marie-Nicole Gras (1980)

Annales de l'institut Fourier

On démontre, à partir de résultats de H.J. Godwin, H. Brunotte et F. Halter-Koch, le théorème suivant : soit K un corps cubique cyclique de conducteur m dont le groupe de Galois G est engendré par σ ; soit E le groupe des unités de norme 1.Soit ϵ ∈ E , ϵ ≠ 1 , telle que 𝒮 ( ϵ ) = 1 2 [ ( ϵ - ϵ σ ) 2 + ( ϵ σ - ϵ σ 2 ) 2 + ( ϵ σ 2 - ϵ ) 2 ] soit minimum. Alors ϵ est un Z [ G ] -générateur de E .

Note on the Hilbert 2-class field tower

Abdelmalek Azizi, Mohamed Mahmoud Chems-Eddin, Abdelkader Zekhnini (2022)

Mathematica Bohemica

Let k be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields 𝕜 = ℚ ( d , - 1 ) , which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

Given an integer n ≥ 3 , let u 1 , ... , u n be pairwise coprime integers ≥ 2 , 𝒟 a family of nonempty proper subsets of { 1 , ... , n } with “enough” elements, and ε a function 𝒟 → { ± 1 } . Does there exist at least one prime q such that q divides ∏ i ∈ I u i - ε ( I ) for some I ∈ 𝒟 , but it does not divide u 1 ⋯ u n ? We answer this question in the positive when the u i are prime powers and ε and 𝒟 are subjected to certain restrictions.We use the result to prove that, if ε 0 ∈ { ± 1 } and A is a set of three or more primes that contains all prime divisors of any number of the form ∏ p ∈ B p - ε 0 for...

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