$3$-rank of ambiguous class groups of cubic Kummer extensions
| العنوان: | $3$-rank of ambiguous class groups of cubic Kummer extensions |
|---|---|
| المؤلفون: | Abdelmalek Azizi, Daniel C. Mayer, Siham Aouissi, Moulay Chrif Ismaili, Mohamed Talbi |
| سنة النشر: | 2018 |
| مصطلحات موضوعية: | Mathematics - Number Theory, Group (mathematics), Root of unity, General Mathematics, 010102 general mathematics, 0211 other engineering and technologies, Order (ring theory), 021107 urban & regional planning, 02 engineering and technology, 01 natural sciences, 11R11, 11R16, 11R20, 11R27, 11R29, 11R37, Cohomology, Combinatorics, Integer, Mathematics::K-Theory and Homology, FOS: Mathematics, Herbrand quotient, Number Theory (math.NT), 0101 mathematics, Abelian group, Unit (ring theory), Mathematics |
| الوصف: | Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(\zeta_3)$ with $d>1$ a cube-free integer and $\zeta_3$ a primitive third root of unity. Denote by $C_{k,3}^{(\sigma)}$ the $3$-group of ambiguous classes of the extension $k/k_0$ with relative group $G=\operatorname{Gal}(k/k_0)=\langle\sigma\rangle$. The aims of this paper are to characterize all extensions $k/k_0$ with cyclic $3$-group of ambiguous classes $C_{k,3}^{(\sigma)}$ of order $3$, to investigate the multiplicity $m(f)$ of the conductors $f$ of these abelian extensions $k/k_0$, and to classify the fields $k$ according to the cohomology of their unit groups $E_{k}$ as Galois modules over $G$. The techniques employed for reaching these goals are relative $3$-genus fields, Hilbert norm residue symbols, quadratic $3$-ring class groups modulo $f$, the Herbrand quotient of $E_{k}$, and central orthogonal idempotents. All theoretical achievements are underpinned by extensive computational results. Comment: 24 pages, 7 tables, 3 figures |
| اللغة: | English |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::032bdedbf603db08f1aff9fbf007500b http://arxiv.org/abs/1804.00767 |
| حقوق: | OPEN |
| رقم الأكسشن: | edsair.doi.dedup.....032bdedbf603db08f1aff9fbf007500b |
| قاعدة البيانات: | OpenAIRE |
| الوصف غير متاح. |